RIGIDITY FOR VON NEUMANN ALGEBRAS OF GRAPH PRODUCT GROUPS II. SUPERRIGIDITY RESULTS

Abstract

In [CDD22], we investigated the structure of $\ast $-isomorphisms between von Neumann algebras $L(\Gamma )$ associated with graph product groups $\Gamma $ of flower-shaped graphs and property (T) wreath-like product vertex groups, as in [CIOS21]. In this follow-up, we continue the structural study of these algebras by establishing that these graph product groups $\Gamma $ are entirely recognizable from the category of all von Neumann algebras arising from an arbitrary nontrivial graph product group with infinite vertex groups. A sharper $C^*$-algebraic version of this statement is also obtained. In the process of proving these results, we also extend the main $W^*$-superrigidity result from [CIOS21] to direct products of property (T) wreath-like product groups.

1 Introduction

In [MvN43], Murray and von Neumann associated in a natural way a von Neumann algebra, denoted by $L(\Gamma )$ , to every countable discrete group $\Gamma $ . Precisely, $L(\Gamma )$ is defined as the weak operator closure of the complex group algebra $\mathbb C[\Gamma ]$ acting by left convolution on the Hilbert space $\ell ^2(\Gamma )$ of square-summable functions on $\Gamma $ . The classification of group von Neumann algebras has since been a central theme in operator algebras, driven by the following fundamental question: What aspects of the group $\Gamma $ are remembered by $L(\Gamma )$ ? This is a challenging problem, as von Neumann algebras tend to forget a lot of the information about the groups from which they were constructed. An excellent illustration of this is Connes’ theorem, which asserts that II $_1$ factors arising from amenable groups are isomorphic to the hyperfinite II $_1$ factor [Co76]. Hence, group von Neumann algebras of infinite conjucacy classes (icc) amenable groups have no memory of the algebraic structure of the underlying group. In sharp contrast, the nonamenable case is far more complex. The emergence of Popa’s deformation/rigidity theory [Po06] has led to the discovery of groups with certain canonical algebraic properties, which are completely retained by their von Neumann algebra. We highlight here only a few of these developments and refer the reader to the surveys [Po06, Va10b, Io12, Io17] for a more complete account in this direction. Popa’s strong rigidity theorem [Po03, Po04] asserts that wreath product groups $\mathbb Z/2\mathbb Z\wr \Gamma $ , where $\Gamma $ is an icc property (T) group, are completely recognizable from the category of all von Neumann algebras arising from arbitrary wreath product groups with Abelian base and icc acting group. Several years later Ioana, Popa and Vaes discovered in [IPV10] the first examples of W $^*$ -superrigid groups, that is, groups that can be entirely reconstructed from their von Neumann algebras. Several additional classes of examples were unveiled subsequently [BV12, Be14, CI17, CD-AD20, CD-AD21, CIOS21]. In this paper, we study superrigidity aspects of von Neumann algebras of graph product groups [Gr90]. These groups are natural generalizations of right-angled Artin and Coxeter groups and play an important role in several subareas of topology and group theory. These groups display a rich structure and have been investigated intensively over the last two decades using deep methods in geometric group theory. In this direction, several landmark results have been discovered (see [HW08, AM10, Wi11, MO13, Ag13]). More recently, certain classes of graph product groups, including many right-angled Artin groups, have been studied through the lens of measured group theory and led to strong rigidity results in the orbit equivalence setting [HH20, HH21].

Graph product groups have also been considered in the analytic framework of von Neumann algebras [CF14, Ca16, CdSS17, DK-E21, CK-E21, CDD22], where various structural and rigidity results of von Neumann algebras of graph product groups have been obtained. In [CDD22], we further developed some of Popa’s powerful deformation/rigidity theoretic methods [Po06] which enabled us to completely describe the structure of all $\ast $ -isomorphisms between von Neumann algebras arising from the fairly large class of graph product groups associated with flower-shaped graphs (see class $\mathrm {CC}_1$ in Definition 1.1 below) and vertex groups which are property (T) wreath-like product groups introduced in [CIOS21]. In essence, our result can be viewed as a von Neumann algebraic counterpart of the group theoretic result of Genevois-Martin [GM19].

This paper continues the study initiated in [CDD22], the main goal being to establish strong rigidity results for the aforementioned graph product groups. For example, we show that any graph product group associated with an asymmetric $\mathrm {CC}_1$ graph and vertex groups that are property (T) wreath-like product groups is completely recognizable from the category of all von Neumann algebras arising from an arbitrary nontrivial graph product group with infinite vertex groups (see Theorem C). Along the way, we also extend the main W $^*$ -superrigidity result from [CIOS21] to direct product groups, consequently obtaining new examples of groups satisfying Connes rigidity conjecture (see Corollary B).

1.1 Statements of the main results

Before stating our results, we recall the construction of graph product groups [Gr90]. Let $\mathscr {G}=(\mathscr V,\mathscr E)$ be a finite graph without loops or multiple edges. The graph product group $\Gamma ={\mathscr {G}}\{\Gamma _v\}$ of a given family of vertex groups $\{\Gamma _v\}_{v\in \mathscr V}$ is the quotient of the free product $\ast _{v\in \mathscr V} \Gamma _v$ by the relations $[\Gamma _u,\Gamma _v]=1$ , whenever $(u,v)\in \mathscr E$ . Note that graph products can be seen as groups that ‘interpolate’ between the direct product $\times _{v\in \mathscr V} \Gamma _v$ (when $\mathscr {G}$ is complete) and the free product $\ast _{v\in \mathscr V} \Gamma _v$ (when $\mathscr {G}$ has no edges). For any subgraph $\mathscr {H}= (\mathscr U,\mathscr F)$ of $\mathscr {G}$ , we denote by $\Gamma _{\mathscr {H}}$ the subgroup generated by $\langle \Gamma _u \,:\,u\in \mathscr U\rangle $ , and we call it the full subgroup of $\mathscr {G}\{\Gamma _v\}$ corresponding to $\mathscr {H}$ . A clique $\mathscr {C}$ of $\mathscr {G}$ is a maximal, complete subgraph of $\mathscr {G}$ . The set of cliques of $\mathscr {G}$ will be denoted by $\mathrm {cliq}(\mathscr {G})$ . The full subgroups $\Gamma _{\mathscr {C}}$ for $\mathscr {C}\in \mathrm {cliq}(\mathscr {G})$ are called the clique subgroups of $\mathscr {G}\{\Gamma _v\}$ .

We are now ready to recall the class of graphs $\mathrm {CC}_1$ that was introduced in [CDD22] and which is used for our main results.

Definition 1.1. A graph $\mathscr {G}$ is called a simple cycle of cliques $(\mathrm {CC}_1)$ if there is an enumeration of its clique set $ \mathrm {cliq}(\mathscr {G})= \{\mathscr {C}_1, ..., \mathscr {C}_n\} $ with $n\geq 4$ , such that the subgraphs $\mathscr {C}_{i,j} :=\mathscr {C}_i \cap \mathscr {C}_j$ satisfy:

$$ \begin{align*} &\mathscr{C}_{i,j}=\begin{cases} \emptyset, \text{ if } \hat i-\hat j\in \mathbb Z_{n} \setminus \{ \hat 1, \widehat {n-1}\}\\ \neq \emptyset, \text{ if } \hat i-\hat j\in \{\hat 1, \widehat {n-1}\} \end{cases} \text{ and }\\ &\mathscr{C}^{\mathrm{int}}_i:=\mathscr{C}_i \setminus \left(\mathscr{C}_{i-1,i} \cup \mathscr{C}_{i,i+1}\right) \neq \emptyset\text{, for all }1\leq k\leq n, \text{ with conventions } 0=n \text{ and } n+1=1. \end{align*} $$

Note, this automatically implies the cardinality $|\mathscr {C}_i |\geq 3$ for all i. Also, such an enumeration $ \mathrm {cliq}(\mathscr {G})= \{\mathscr {C}_1,..., \mathscr {C}_n\} $ is called a consecutive clique’s enumeration.

Example 1.2. A basic example of such a graph is any simple, length n, cycle of triangles $\mathscr F_n=(\mathscr V_n,\mathscr E_n)$ , which essentially looks like a flower-shaped graph with n petals:

(1.1)

In fact, any graph from $\mathrm {CC}_1$ is a two-level clustered graph that is a specific retraction of $\mathscr F_n$ ; for more details, the reader may consult [CDD22, Section 2].

In this paper, we investigate various superrigidity aspects for group von Neumann algebras of the aforementioned graph product groups. Since the underlying groups are essentially built-up from collections of large clique groups that are just direct product groups, it is natural to first tackle the superrigidity question for these types of groups. One could think of this as being the degenerate case. In this direction, we were able to establish a product rigidity result for property (T) wreath-like product groups in the same spirit with [CdSS15, Theorem A] or the more recent results [CD-AD20, Dr20]. Specifically, we have the following:

Theorem A. For every $1\leq k\leq n$ , let $\Gamma _k\in \mathcal W\mathcal R (A_k, B_k \curvearrowright I_k)$ be property (T) groups where $A_k$ is Abelian, $B_k$ is an icc subgroup of a hyperbolic group, $B_k \curvearrowright I_k$ has amenable stabilizers and denote $\Gamma = \Gamma _1\times \dots \times \Gamma _n$ .

Assume that $t>0$ is a scalar and $\Lambda $ is an arbitrary group satisfying ${\mathcal M}={\mathcal L}(\Gamma )^t={\mathcal L}(\Lambda )$ .

Then one can find a direct product decomposition $\Lambda = \Lambda _1\times \dots \times \Lambda _n$ , some scalars $t_1,\dots ,t_n>0$ with $t_1\cdots t_n=t$ and a unitary $u\in {\mathcal M}$ satisfying ${\mathcal L}(\Gamma _{i})^{t_i}=u{\mathcal L}(\Lambda _i) u^*$ , for any $1\leq i \leq n.$

This theorem in conjunction with the W $^*$ -superrigidity result from [CIOS21, Theorem 9.9] immediately yields that essentially all finite direct products of property (T) wreath-like product groups covered by the prior theorem are completely recognizable from their von Neumann algebras. Before we introduce the result, we recall the notion of group-like $*$ -isomorphism between von Neumann algebras. Let $\Gamma $ and $\Lambda $ be countable groups. Let $\eta :\Gamma \rightarrow \mathbb T$ be a multiplicative character and $\delta : \Gamma \rightarrow \Lambda $ a group isomorphism. Consider the group von Neumann algebras ${\mathcal L}(\Gamma )$ , and ${\mathcal L}(\Lambda )$ and denote by $\{u_g \,:\, g\in \Gamma \}$ and $\{v_\lambda \,:\, \lambda \in \Lambda \}$ their corresponding canonical group unitaries. Then the map $\Gamma \ni u_g\rightarrow \eta (g)v_{\delta (g)}\in \Lambda $ canonically extends to a $\ast $ -isomorphism denoted by $\Psi _{\eta ,\delta }: {\mathcal L}(\Gamma )\rightarrow {\mathcal L}(\Lambda )$ .

Corollary B. For every $1\leq k\leq n$ , let $\Gamma _k\in \mathcal W\mathcal R (A_k, B_k \curvearrowright I_k)$ be property (T) groups where $A_k$ is Abelian, $B_k$ is an icc subgroup of a hyperbolic group, $B_k \curvearrowright I_k$ has amenable stabilizers and the set $\{i\in I \; | \; g\cdot i\neq i\}$ is infinite for any $g\in B_k\setminus \{1\}$ . Denote $\Gamma = \Gamma _1\times \dots \times \Gamma _n$ .

Let $t>0$ be any scalar, $\Lambda $ be an arbitrary group and $\theta : {\mathcal L}(\Gamma )^t\rightarrow {\mathcal L}(\Lambda )$ be any $\ast $ -isomorphism.

Then $t=1$ , and one can find a character $\eta :\Gamma \to \mathbb T$ , a group isomorphism $\delta :\Gamma \to \Lambda $ and a unitary $u\in {\mathcal L}(\Lambda )$ satisfying $\theta =\mathrm {ad}(u)\circ \Psi _{\eta ,\delta }$ .

We mention in passing that since property (T) passes to (finite) direct products of groups, Corollary B provides new examples of property (T) groups which satisfy Popa’s strengthening of Connes rigidity conjecture [Co82, Po06]. In particular, the result also shows that these property (T) factors have trivial fundamental group and also verify Jones’s outer automorphism problem [Jo00], providing additional examples to recent similar results [CDHK20, CIOS21, CIOS23].

To tackle the problem of reconstructing the above graph product groups $\Gamma $ from their factors ${\mathcal L}(\Gamma )$ , we develop new aspects of a more conceptual principle from [CI17, CD-AD20] called peripheral reconstruction $W^*$ -method; this consists of exploiting the natural tension that occurs between ‘a peripheral structure’ and a ‘direct product structure’ in the group. In our specific situation, this means that if $\Lambda $ is any group, such that $\mathcal L(\Gamma )=\mathcal L(\Lambda )$ , then the main strategy is to first identify in $\Lambda $ collections of subgroups that play the same role as the ‘peripheral structure’ of $\Gamma $ given by its full subgroups associated to the cliques in the underlying graph.

In this direction, using an approach combining the comultiplication map [Io10, IPV10, PV11], the ultrapower methods from [Io11] (see also [DHI16, CdSS15, KV15]), a method for reconstructing malnormal groups from [CD-AD20, CI17] and Corollary B, we are able to show that the clique subgroups of our graph product groups are, in fact, completely recognizable under the W $^*$ -equivalence. For the precise statement, the reader may consult Theorem 7.1. While this result does not show complete reconstruction of these graph products from the category of all group von Neumann algebras, it can be used effectively to show these groups are recognizable from the von Neumann algebras associated with an extensive family of groups—all nontrivial graph products with infinite vertex groups.

Theorem C. Let $\mathscr {G}\in \mathrm {CC}_1$ , let $\mathrm {cliq}(\mathscr {G})=\{ \mathscr {C}_1, \ldots , \mathscr {C}_n\}$ be its consecutive cliques enumeration and assume that $|\mathscr {C}_i|\neq |\mathscr {C}_j|$ whenever $i\neq j$ . Let $\Gamma = \mathscr {G}\{\Gamma _v\}$ be a graph product group where all vertex groups $\Gamma _v$ are property (T) wreath-like product groups $\Gamma _v\in \mathcal W\mathcal R(A_v,B_v)$ with $A_v$ Abelian and $B_v$ an icc subgroup of a hyperbolic group.

Then, for any nontrivial graph product group $\Lambda $ with infinite vertex groups satisfying ${\mathcal L}(\Gamma )\cong {\mathcal L}(\Lambda )$ , we have $\Gamma \cong \Lambda $ .

In fact, we have obtained a more precise version of the above result (see Theorem 7.6). Not only can we derive an isomorphism of the underlying groups, but we can, in fact, completely describe all isomorphisms between ${\mathcal L}(\Gamma )$ and ${\mathcal L}(\Lambda )$ . Namely, they appear as compositions between the canonical group-like isomorphisms $\Psi _{\eta , \delta }$ induced by a group isomorphism $\delta :\Gamma \to \Lambda $ , a character $\eta :\Gamma \to \mathbb T$ and the local automorphisms of graph product group von Neumann algebras introduced in [CDD22] (see also Section 2.3).

The result yields new applications towards rigidity in the $C^*$ -algebraic framework. Since these graph product groups have trivial amenable radical (see [CDD22, Lemma 4.3]) using [BKKO14, Theorem 1.3], it follows that their reduced C $^*$ -algebras have unique trace. Therefore, the conclusion of Theorem C still holds true for reduced group $C^*$ -algebras. Moreover, if one assumes, in addition, these groups are torsion free, then the previous theorem actually yields a formally stronger reconstruction statement allowing us to drop the hypothesis assumption that the vertex groups of the target $\Lambda $ are infinite.

Corollary D. Let $\mathscr {G}\in \mathrm {CC}_1$ , let $\mathrm {cliq}(\mathscr {G})=\{ \mathscr {C}_1,\ldots , \mathscr {C}_n\}$ be a consecutive clique’s enumeration and assume that $|\mathscr {C}_i|\neq |\mathscr {C}_j|$ whenever $i\neq j$ . Let $\Gamma = \mathscr {G}\{\Gamma _v\}$ be a graph product group where all vertex groups $\Gamma _v$ are torsion free, property (T) wreath-like product groups $\Gamma _v\in \mathcal W\mathcal R(A_v,B_v)$ with $A_v$ Abelian and $B_v$ an icc subgroup of a hyperbolic group.

Then, for any nontrivial graph product group $\Lambda $ satisfying $C^*_r(\Gamma )\cong C^*_r(\Lambda )$ , we have $\Gamma \cong \Lambda $ .

To see this, just notice that the graph product groups $\Gamma $ covered by this corollary are torsion free and also satisfy the Baum-Connes conjecture (see Proposition 2.2, [O-O01b, Theorem 7.7] and [MY02, Theorem 20]). Hence, their reduced $C^*$ -algebras $C^*_r(\Gamma )$ are projectionless and so are $C^*_r(\Lambda )$ . This further entails that $\Lambda $ is also torsion free; in particular, the vertex subgroups of $\Lambda $ are automatically infinite. Thus, the conclusion follows from Theorem C.

2 Preliminaries

2.1 Terminology

Throughout this article, all von Neumann algebras are denoted by calligraphic letters, for example, ${\mathcal M}$ , $\mathcal N$ , $\mathcal P$ , $\mathcal {Q}$ , etc. All von Neumann algebras ${\mathcal M}$ will be tracial, that is, endowed with a unital, faithful, normal linear functional $\tau :{\mathcal M}\rightarrow \mathbb C$ satisfying $\tau (xy)=\tau (yx)$ , for all $x,y\in {\mathcal M}$ . This induces a norm on ${\mathcal M}$ given by the formula $\|x\|_2=\tau (x^*x)^{1/2}$ , for any $x\in {\mathcal M}$ . The $\|\cdot \|_2$ -completion of ${\mathcal M}$ will be denoted by $L^2({\mathcal M})$ . Given a von Neumann algebra ${\mathcal M}$ , we will denote by $\mathscr U({\mathcal M})$ its unitary group, by $\mathcal P({\mathcal M})$ its projections set and by $\mathcal Z({\mathcal M})$ its center. Given a unital inclusion ${\mathcal N}\subset {\mathcal M}$ of von Neumann algebras, we denote by $E_{\mathcal N}:{\mathcal M}\to \mathcal N$ the unique $\tau $ -preserving conditional expectation from ${\mathcal M}$ onto $\mathcal N$ , by $e_{\mathcal N}:L^2({\mathcal M})\to L^2(\mathcal N)$ the orthogonal projection onto $L^2(\mathcal N)$ and by $\langle {\mathcal M},e_{\mathcal {N}}\rangle $ the Jones’s basic construction of $\mathcal N\subset {\mathcal M}$ . Also, we denote by $\mathcal N'\cap {\mathcal M} =\{ x\in {\mathcal M} \,:\, [x, \mathcal N]=0\}$ the relative commmutant of $\mathcal N$ inside ${\mathcal M}$ and by $\mathscr N_{\mathcal M}(\mathcal N)=\{ u\in \mathscr U({\mathcal M})\,:\, u\mathcal N u^*=\mathcal N\}$ the normalizer of $\mathcal N$ inside ${\mathcal M}$ . We say that the inclusion $\mathcal N$ is regular in ${\mathcal M}$ if $\mathscr N_{{\mathcal M}}(\mathcal N)"={\mathcal M}$ and irreducible if $\mathcal N'\cap {\mathcal M}=\mathbb C 1$ .

For a group inclusion $\Sigma < \Gamma $ , we denote by $C_{\Gamma }(\Sigma )=\{ g\in \Gamma \,|\, [g,\Sigma ]=1\}$ its centralizer in $\Gamma $ and by $vC_{\Gamma }(\Sigma )=\{ g\in \Gamma \,|\, |g^\Sigma |<\infty \}$ its virtual centralizer. Note that $vC_\Gamma (\Gamma )=1$ precisely when $\Gamma $ is icc. We denote by $N_\Gamma (\Sigma )=\{ g\in \Gamma \,|\, g\Sigma g^{-1}=\Sigma \}$ the normalizer of $\Sigma $ in $\Gamma $ .

2.2 Graph product groups

We now recall the notion of graph product groups introduced by Green [Gr90] while also highlighting some of their properties that are relevant to our work. Let $\mathscr {G}=(\mathscr V,\mathscr E)$ be a finite simple graph, where $\mathscr V$ and $\mathscr E$ denote its vertex and edge sets, respectively. Let $\{\Gamma _v\}_{v\in \mathscr V}$ be a family of groups called vertex groups. The graph product group associated with these data, denoted by $\mathscr {G}\{\Gamma _v,v \in \mathscr V\}$ or simply $\mathscr {G}\{\Gamma _v\}$ , is the group generated by $\Gamma _v$ , $v\in \mathscr V$ with the only relations being $[\Gamma _u, \Gamma _v] = 1$ , whenever $(u,v)\in \mathscr E$ . Given any subset $\mathscr U\subset \mathscr V$ , the subgroup $\Gamma _{\mathscr {U}} =\langle \Gamma _u \,:\,u\in \mathscr {U}\rangle $ of $\mathscr {G}\{\Gamma _v,v\in \mathscr V\}$ is called a full subgroup. This can be identified with the graph product $\mathscr {G}_{\mathscr {U}}\{\Gamma _u,u \in \mathscr {U}\}$ corresponding to the subgraph $\mathscr {G}_{\mathscr {U}}$ of $\mathscr {G}$ , spanned by the vertices of $\mathscr {U}$ . For every $v \in \mathscr V$ , we denote by $\mathrm {lk}(v)$ the subset of vertices $w\neq v$ so that $(w,v)\in \mathscr E$ . Similarly, for every $\mathscr {U} \subseteq \mathscr V$ , we denote by ${\mathrm {lk}(\mathscr {U}) = \cap _{u\in \mathscr {U}}\mathrm {lk}(u)}$ . Also, we make the convention that $\mathrm {lk}(\emptyset ) = \mathscr V$ . Notice that $\mathscr {U} \cap \mathrm {lk}(\mathscr {U}) = \emptyset $ .

Graph product groups naturally admit many amalgamated free product decompositions as follows (see [Gr90, Lemma 3.20]). For any $w \in \mathscr V$ , we have

(2.1) $$ \begin{align}\mathscr{G}\{\Gamma_v\} = \Gamma_{\mathscr V\setminus \{w\}} \ast_{ \Gamma_{\mathrm{lk}}(w)} \Gamma_{\mathrm{st}(w)},\end{align} $$

where $\mathrm {st} (w) = \{w\} \cup \mathrm {lk} (w)$ . Notice that $\Gamma _{\mathrm {lk}(w)}\lneqq \Gamma _{\mathrm {st}(w)}$ , but it could be the case that $\Gamma _{\mathrm {lk}(w)}=\Gamma _{\mathscr V\setminus \{w\}} $ , when $\mathscr V=\mathrm {st}(w)$ . In this case, the amalgam decomposition is called degenerate.

Similarly, for every subgraph $\mathscr {U}\subset \mathscr {G}$ , we denote by $\mathrm {st}(\mathscr {U})= \mathscr {U} \cup \mathrm {lk}(\mathscr {U} )$ . A maximal complete subgraph $\mathscr {C}\subseteq \mathscr {G}$ is called a clique, and the collections of all cliques of $\mathscr {G}$ will be denoted by $\mathrm {cliq}(\mathscr {G})$ . Below, we highlight various properties of full subgroups that will be useful in this paper.

Proposition 2.1. Let $\Gamma =\mathscr {G} \{\Gamma _v\}$ be any graph product of groups, $g\in \Gamma $ and let $\mathscr {S},\mathscr {T}\subseteq \mathscr {G}$ be any subgraphs. Then the following hold.

1. 1. [AM10, Lemma 3.9] If $g \Gamma _{\mathscr {T}} g^{-1}\subset \Gamma _{\mathscr {T}}$ , then $g \Gamma _{\mathscr {T}} g^{-1}=\Gamma _{\mathscr {T}}$ .

2. 2. [AM10, Proposition 3.13] $N_{\Gamma }(\Gamma _{\mathscr {T}})=\Gamma _{\mathscr {T}\cup \mathrm {link}(\mathscr {T})}$ .

3. 3. [AM10, Proposition 3.4] There exist $\mathscr {D}\subseteq \mathscr {S}\cap \mathscr {T}$ and $h\in \Gamma _{\mathscr {T}}$ , such that $g \Gamma _{\mathscr {S}} g^{-1}\cap \Gamma _{\mathscr {T}}=h \Gamma _{\mathscr {D}} h^{-1}$ .

For further use, we also record the following easy consequence of results in [O-O01b, O-O98].

Proposition 2.2. Let $\Gamma =\mathscr {G} \{\Gamma _v\}$ be any graph product of groups. Then the following hold:

1. 1. If $\Gamma _v$ is torsion free for all $v\in \mathscr V$ , then $\Gamma $ is also torsion free;

2. 2. If $\Gamma _v$ is torsion free and satisfies Baum-Connes conjecture for all $v\in \mathscr V$ , then so is $\Gamma $ .

Proof. Torsion free passes to both direct products and free product with amalgamation. Thus, using either product decomposition or (2.1), the statement follows by induction on the number of vertices in $\mathscr {G}$ .

To see part 2, notice that the class of torsion free groups that satisfies Baum-Connes property is closed under taking finite direct products [O-O01b] and also undertaking free product with amalgamation [O-O98]. Once again, using either product decomposition or (2.1), the statement follows by induction on the number of vertices in $\mathscr {G}$ .

2.3 Cycles of cliques graphs and their von Neumann algebras

In the first part of this section, we recall from [CDD22, Section 7] a canonical family of $*$ -isomorphisms between graph product group von Neumann algebras when the underlying graphs belong to $\mathrm { CC}_1$ . Let $\mathscr {G},\mathscr {H} \in \mathrm {CC}_1$ be isomorphic graphs, and fix $\sigma : \mathscr {G} \rightarrow \mathscr {H}$ an isometry. Let $\mathrm {cliq}( \mathscr {G}) =\{ \mathscr {C}_1, \ldots ,\mathscr {C}_n\}$ be an enumeration of consecutive cliques. Let $\Gamma _{\mathscr {G}}$ and $\Lambda _{\mathscr {H}}$ be graph product groups, and assume that for every $1\leq i\leq n $ , there are $\ast $ -isomorphisms $\theta _{i-1,i}: {\mathcal L}(\Gamma _{\mathscr {C}_{i-1,i}})\rightarrow \mathcal L(\Lambda _{\mathscr {C}_{\sigma (\mathscr {C}_{i-1,i})}}) $ , $\xi _{i}:\mathcal L(\Gamma _{\mathscr {C}^{\mathrm {int}}_i})\rightarrow \mathcal L(\Lambda _{\sigma (\mathscr {C}^{\mathrm { int}}_i)})$ and $\theta _{i,i+1}:{\mathcal L}(\Gamma _{\mathscr {C}_{i,i+1}})\rightarrow {\mathcal L}(\Lambda _{\mathscr {C}_{\sigma (\mathscr {C}_{i,i+1})}}) $ ; here, and in what follows, we convene as before that $n=0$ and $n+1=1$ . By [CDD22, Theorem 7.1] (see also [CF14]), these $\ast $ -isomorphisms induce a unique $\ast $ -isomorphism $\phi _{\theta ,\xi , \sigma }:{\mathcal L}(\Gamma _{\mathscr {G}})\rightarrow {\mathcal L}(\Lambda _{\mathscr {H}})$ defined as

(2.2) $$ \begin{align} \phi_{\theta,\xi, \sigma}(x)=\begin{cases} \theta_{i-1,i}(x), \text{ if } x\in {\mathcal L}(\Gamma_{\mathscr{C}_{i-1,i}})\\ \xi_i(x), \qquad \text{if } x\in {\mathcal L}(\Gamma_{\mathscr{C}^{\mathrm{int}}_i }) \end{cases} \end{align} $$

for all $1\leq i\leq n$ .

When $\Gamma _{\mathscr {G}} =\Lambda _{\mathscr {H}}$ , this construction yields a group of $\ast $ -automorphisms of ${\mathcal L}(\Gamma _{\mathscr {G}})$ , denoted by $\mathrm {Loc}_{\mathrm {c,g}}({\mathcal L}(\Gamma _{\mathscr {G}}))$ . We also denote by $\mathrm {Loc}_{\mathrm {c}}({\mathcal L}(\Gamma _{\mathscr {G}}))$ the subgroup of all local automorphisms satisfying $\sigma =\mathrm {Id}$ . Next, we highlight a class of automorphisms in $\mathrm {Loc}_{\mathrm {c}}({\mathcal L}(\Gamma _{\mathscr {G}}))$ needed to state Theorem 7.6. Consider n-tuples $a =(a_{i,i+1})_i$ and $b= (b_i)_i$ of nontrivial unitaries $a_{i,i+1} \in \mathcal L(\Gamma _{\mathscr {C}_{i-1,i}})$ and $b_i\in \mathcal L(\Gamma _{\mathscr {C}^{\mathrm {int}}_i})$ , for every $1\leq i\leq n$ . If in (2.2) we let $\theta _{i,i+1}= \mathrm {ad} (a_{i,i+1}) $ and $\xi _i =\mathrm {ad} (b_i)$ , then the corresponding local automorphism $\phi _{\theta ,\xi , \mathrm {Id}}$ is, in general, an outer automorphism of $\mathcal L(\Gamma )$ (see [CDD22, Proposition 7.4]) and will be denoted by $\phi _{a,b}$ throughout the paper.

We conclude this subsection by recording an important result from [CDD22] that is essential for deriving Theorem C.

Theorem 2.3 [CDD22, Theorem 5.2].

Let $\mathscr {G}=\{\mathscr V,\mathscr E\}$ be a graph in the class $\mathrm {CC}_1$ , and let $\mathscr {C}_1,...,\mathscr {C}_n$ be an enumeration of its consecutive cliques. Let { $\Gamma _v|\;v\in \mathscr V\}$ be a collection of icc groups, and let $\Gamma _{\mathscr {G}}$ be the corresponding graph product group. For each $1\leq i\leq n$ , let $a_{i,i+1}\in \mathscr {U}({\mathcal L}(\Gamma _{\mathscr {C}_{i,i+1}}))$ , $b_{i,i+1}\in \mathscr {U}(\mathcal L(\Gamma _{\mathscr {C}_i \cup \mathscr {C}_{i+1}\setminus \mathscr {C}_{i,i+1}})$ and denote $x_{i,i+1}= a_{i,i+1} b_{i,i+1}$ .

If $x_{1,2} x_{2,3}\cdots x_{n-1,n}x_{n,1}=1$ , then for each $1\leq i\leq n $ , one can find $a_i \in \mathscr {U}({\mathcal L}(\Gamma _{\mathscr {C}_{i-1,i}}))$ , $b_i \in \mathscr {U}({\mathcal L}(\Gamma _{\mathscr {C}^{\mathrm {int}}_i}))$ and $c_i \in \mathscr {U}(\mathcal L (\Gamma _{\mathscr {C}_{i,i+1}}))$ so that $x_{i,i+1}= a_i b_i c_i b^*_{i+1} a^*_{i+2}c^*_{i+1}$ . Here, we convene that $n+1=1$ , $n+2=2$ , etc.

2.4 Wreath-like product groups

In [CIOS21], a new category of groups called wreath-like product groups was introduced. To recall their construction, let A and B be any countable groups, and let $B \curvearrowright I$ be an action on a countable set. One says W is a wreath-like product of A and $B\curvearrowright I$ if it can be realized as a group extension

(2.3) $$ \begin{align} 1 \rightarrow \bigoplus_{i\in I} A_i\hookrightarrow W \overset{\varepsilon}{\twoheadrightarrow} B \rightarrow 1 \end{align} $$

which satisfies the following properties:

1. a) $A_i\cong A$ for all $i\in I$ , and

2. b) the action by conjugation of W on $\bigoplus _{i\in I} A_i$ permutes the direct summands according to the rule

   $$ \begin{align*}w A_i w^{-1}= A_{\varepsilon(w)i}\text{ for all }w\in W, i\in I.\end{align*} $$

The class of all such wreath-like groups is denoted by $\mathcal W\mathcal R(A, B\curvearrowright I)$ . When $I= B$ and the action $B\curvearrowright I$ is by translation, this consists of so-called regular wreath-like product groups, and we simply denote their class by $\mathcal W\mathcal R(A, B)$ .

Notice that every classical generalized wreath product $A\wr _I B \in \mathcal W\mathcal R (A,B\curvearrowright I)$ . However, building examples of nonsplit wreath-like products is far more involved. Indeed, using deep methods in group theoretic Dehn filling and Cohen-Lyndon subgroups, it was shown in [CIOS21] that large classes of such wreath-like products exist, including many with property (T).

Theorem 2.4 [CIOS21].

Let G be a hyperbolic group. For every finitely generated group A, there exists a quotient W of G, such that $W\in \mathcal W\mathcal R (A,B)$ for some hyperbolic group B. In particular, when G has property (T) (e.g. any uniform lattice in $Sp(n,1)$ , $n\geq 2$ ), then so does W.

2.5 Popa’s intertwining-by-bimodules techniques

We next recall from [Po03, Theorem 2.1, Corollary 2.3] Popa’s intertwining-by-bimodules technique. Let $\mathcal {Q}\subset {\mathcal M}$ be a von Neumann subalgebra. Jones’s basic construction $\langle {\mathcal M},e_{\mathcal {Q}}\rangle $ is defined as the von Neumann subalgebra of $\mathbb B(L^2({\mathcal M}))$ generated by ${\mathcal M}$ and the orthogonal projection $e_{\mathcal {Q}}$ from $L^2({\mathcal M})$ onto $L^2(\mathcal {Q})$ . The basic construction $\langle {\mathcal M},e_{\mathcal {Q}}\rangle $ has a faithful semi-finite trace given by $\text {Tr}(xe_{\mathcal {Q}} y)=\tau (xy)$ , for every $x,y\in {\mathcal M}$ . We denote by $L^2(\langle {\mathcal M},e_{\mathcal {Q}}\rangle )$ the associated Hilbert space and endow it with the natural ${\mathcal M}$ -bimodule structure.

Theorem 2.5 [Po03].

Let $({\mathcal M},\tau )$ be a tracial von Neumann algebra and $\mathcal P\subset p{\mathcal M} p, \mathcal {Q}\subset q{\mathcal M} q$ be von Neumann subalgebras. Then the following are equivalent:

1. 1. There exist projections $p_0\in \mathcal P, q_0\in \mathcal {Q}$ , a $*$ -homomorphism $\theta :p_0\mathcal P p_0\rightarrow q_0\mathcal {Q} q_0$ and a nonzero partial isometry $v\in q_0{\mathcal M} p_0$ , such that $\theta (x)v=vx$ , for all $x\in p_0\mathcal P p_0$ .

2. 2. There is no sequence $(u_n)_{n\ge 1}\subset \mathcal U(\mathcal P)$ satisfying $\|E_{\mathcal {Q}}(x^*u_ny)\|_2\rightarrow 0$ , for all $x,y\in p{\mathcal M}$ .

If one of these equivalent conditions holds true, we write $\mathcal P\prec _{{\mathcal M}}\mathcal {Q}$ , and say that a corner of $\mathcal P$ embeds into $\mathcal {Q}$ inside ${\mathcal M}$ . Moreover, if $\mathcal P p'\prec _{{\mathcal M}}\mathcal {Q}$ for any nonzero projection $p'\in \mathcal P'\cap p{\mathcal M} p$ , then we write $\mathcal P\prec ^{s}_{{\mathcal M}}\mathcal {Q}$ .

We continue by recording several elementary facts about intertwining results in group von Neumann algebras (of graph product groups).

Lemma 2.6 [CI17, Lemma 2.2].

Let $\Gamma _1, \Gamma _2 < \Gamma $ be countable groups, such that ${\mathcal L}(\Gamma _1) \prec _{{\mathcal L}(\Gamma )} {\mathcal L}(\Gamma _2)$ . Then one can find $g \in \Gamma $ , such that $[\Gamma _1 : \Gamma _1 \cap g \Gamma _2g^{-1}] < \infty $ .

Corollary 2.7 [CDD22, Lemma 2.3].

Let $\Gamma =\mathscr {G} \{\Gamma _v\}$ be any graph product of infinite groups, and let $\mathscr {S},\mathscr {T}\subseteq \mathscr {G}$ be any subgraphs. If ${\mathcal L}(\Gamma _{\mathscr {S}})\prec _{{\mathcal L}(\Gamma )} \mathcal L(\Gamma _{\mathscr {T}})$ , then $\mathscr {S}\subset \mathscr {T}$ .

We also recall the following result, which classifies all rigid subalgebras of von Neumann algebras associated to graph product groups.

Theorem 2.8 [CDD22, Theorem 6.1].

Let $\Gamma = \mathscr {G}\{\Gamma _v\}$ be a graph product group, let $\Gamma \curvearrowright \mathcal P$ be any trace preserving action and denote by ${\mathcal M}=\mathcal P\rtimes \Gamma $ the corresponding crossed product von Neumann algebra. Let $r\in {\mathcal M}$ be a projection, and let $\mathcal {Q}\subset r{\mathcal M} r$ be a property (T) von Neumann subalgebra.

Then one can find a clique $\mathscr {C}\in \mathrm {cliq}(\mathscr {G})$ , such that $\mathcal {Q}\prec _{\mathcal M} \mathcal P\rtimes \Gamma _{\mathscr {C}}$ . Moreover, if $Q\nprec \mathcal P \rtimes \Gamma _{\mathscr {C} \setminus \{c\}}$ for all $c\in \mathscr {C}$ , then one can find projections $q\in \mathcal {Q}$ , $q'\in \mathcal {Q}'\cap r{\mathcal M} r$ with $qq'\neq 0$ and a unitary $u\in {\mathcal M}$ , such that $u q\mathcal {Q} q q'u^{*}\subseteq \mathcal P \rtimes \Gamma _{\mathscr {C}}$ . In particular, if $\mathcal P\rtimes \Gamma _{\mathscr {C}}$ is a factor, then one can take $q=1$ above.

The following result is a direct corollary of [IPP05, Theorem 1.2.1]. For completeness, we provide all the details.

Corollary 2.9. Let $\Gamma =\mathscr {G} \{\Gamma _v\}$ be any graph product of infinite groups, and let ${\mathcal M}={\mathcal L}(\Gamma )$ . Let $\mathcal A\subset {\mathcal L}(\Gamma _v)$ be a diffuse von Neumann subalgebra, for some $v\in \mathscr {G}$ . Then $\mathcal A '\cap {\mathcal M} = {\mathcal L}(\Gamma _{\mathrm {link}(v)})\bar \otimes (\mathcal A'\cap {\mathcal L}(\Gamma _v))$ .

Proof. From definitions, we have $\mathcal A '\cap {\mathcal M} \supset {\mathcal L}(\Gamma _{\mathrm {link}(v)})\bar \otimes (\mathcal A'\cap {\mathcal L}(\Gamma _v))$ . For proving the reverse containment, we note that we can write ${\mathcal M}$ as an amalgamated free product ${{\mathcal M}= {\mathcal L}(\Gamma _{\mathrm {star} (v)})\ast _{{\mathcal L}(\Gamma _{\mathrm {link} (v)})} {\mathcal L}(\Gamma _{\mathscr {G}\setminus \{v\}})}$ . Since $\mathcal A$ is diffuse, it follows that $\mathcal A\nprec _{{\mathcal L}(\Gamma _{\mathrm {star} (v)})} {\mathcal L}(\Gamma _{\mathrm {link} (v)}) $ . By [IPP05, Theorem 1.2.1], it follows that $\mathcal A '\cap {\mathcal M}\subset \mathcal A '\cap {\mathcal L}(\Gamma _{\mathrm {star} (v)}) ={\mathcal L}(\Gamma _{\mathrm {link}(v)})\bar \otimes (\mathcal A'\cap {\mathcal L}(\Gamma _v)).$

Lemma 2.10. Let $\Sigma <\Gamma $ be countable groups, and denote ${\mathcal M}={\mathcal L} (\Gamma )$ . Assume that ${\mathcal P\subset p{\mathcal M} p}$ and $\mathcal {Q}\subset q{\mathcal M} q$ are von Neumann subalgebras satisfying $\mathcal P\prec _{{\mathcal M}} {\mathcal L}(\Sigma )$ and ${\mathcal L}(\Sigma )\prec _{{\mathcal M}} \mathcal {Q}$ .

If $\mathcal P'\cap p{\mathcal M} p$ is amenable, then $\mathcal {Q} '\cap q{\mathcal M} q$ has an amenable direct summand.

Proof. Note that one can find an increasing sequence of groups $\Omega _1 \leqslant \Omega _{2}\leqslant \cdots \leqslant \mathrm {vC}_\Gamma (\Sigma )$ normalized by $\Sigma $ with $\cup _{n\ge 1} \Omega _n=\mathrm { vC}_\Gamma (\Sigma )$ whose centralizers form a descending sequence $\Sigma \geqslant C_\Sigma (\Omega _1)\geqslant \cdots \geqslant C_\Sigma (\Omega _n )\geqslant C_\Sigma (\Omega _{n+1})\geqslant \cdots $ of finite index subgroups. Indeed, recall that $\mathrm {vC}_\Gamma (\Sigma )=\{g\in \Gamma |\; g^\Sigma \text { is finite} \}$ , and let $\{\mathcal O_n\}_{n\ge 1}$ be a countable enumeration of all the finite orbits of the action by conjugation of $\Sigma $ on on $\Gamma $ . Note that $\Omega _n:=\langle \cup _{1\leq k\leq n} \mathcal O_k \rangle $ satisfies the assumption.

Next, since $[\Sigma : C_\Sigma (\Omega _n )]<\infty $ , we get from the assumption that $\mathcal P\prec _{{\mathcal M}} {\mathcal L}(C_\Sigma (\Omega _n ))$ , for all $n\ge 1$ . By passing to relative commutants, we derive from [Va08, Lemma 3.5] that ${\mathcal L}(\Omega _n)\prec _{{\mathcal M}} \mathcal P'\cap p{\mathcal M} p$ , which implies that $L(\Omega _n)$ has an amenable direct summand for any $n\ge 1$ . This further implies that there exists a nonzero projection $z_n\in \mathcal Z(L(\Omega _n))$ , such that $L(\Omega _n)z_n$ is amenable (see [Io12b, Remark 2.2]). Standard arguments now imply that $\Omega _n$ is amenable for any $n\ge 1$ . This shows that $\mathrm {vC}_\Gamma (\Sigma )$ is amenable. Finally, since ${\mathcal L}(\Sigma )\prec _{{\mathcal M}} \mathcal {Q}$ , we pass to relative commutants and apply [Va08, Lemma 3.5] to get that $\mathcal {Q}'\cap q{\mathcal M} q\prec _{{\mathcal M}} {\mathcal L}(\mathrm {vC}_\Gamma (\Sigma ))$ . Since $\mathrm {vC}_\Gamma (\Sigma )$ is amenable, the conclusion follows.

2.6 Relative amenability

A tracial von Neumann algebra $({\mathcal M},\tau )$ is amenable if there exists a positive linear functional $\Phi :\mathbb B(L^2({\mathcal M}))\to \mathbb C$ satisfying $\Phi _{|{\mathcal M}}=\tau $ and $\Phi $ is ${\mathcal M}$ -central, meaning $\Phi (xT)=\Phi (Tx),$ for all $x\in {\mathcal M}$ and $T\in \mathbb B(L^2({\mathcal M}))$ . By Connes’s celebrated theorem [Co76], it follows that ${\mathcal M}$ is amenable if and only if ${\mathcal M}$ is approximately finite dimensional.

We continue by recalling the notion of relative amenability introduced by Ozawa and Popa in [OP07]. Let $({\mathcal M},\tau )$ be a tracial von Neumann algebra. Let $p\in {\mathcal M}$ be a projection and $\mathcal P\subset p{\mathcal M} p,\mathcal {Q}\subset {\mathcal M}$ be von Neumann subalgebras. Following [OP07, Definition 2.2], we say that $\mathcal P$ is amenable relative to $\mathcal {Q}$ inside ${\mathcal M}$ if there exists a positive linear functional $\Phi :p\langle {\mathcal M},e_{\mathcal {Q}} \rangle p\to \mathbb C$ , such that $\Phi _{|p{\mathcal M} p}=\tau $ and $\Phi $ is $\mathcal P$ -central. We say that $\mathcal P$ is strongly nonamenable relative to $\mathcal {Q}$ inside ${\mathcal M}$ if $\mathcal P p'$ is nonamenable relative to $\mathcal {Q}$ inside ${\mathcal M}$ for any nonzero projection $p'\in \mathcal P '\cap p{\mathcal M} p$ .

In this section, we prove a result (Proposition 2.12) that is inspired by [PV12, Proposition 3.2]; its proof is similar to the approach from [Dr19a, Lemma 2.10]. First, we show the following well-known result, which computes the basic construction of natural inclusions of subalgebras arising from trace preserving actions of countable groups (see also [Be14, Lemma 2.5]).

Lemma 2.11. Let $\Gamma \curvearrowright {\mathcal D}$ be a trace preserving action of a countable group. Let $\Sigma <\Gamma $ be a subgroup, and denote ${\mathcal M}={\mathcal D}\rtimes \Gamma $ and $\mathcal N={\mathcal D}\rtimes \Sigma $ . Denote by $\mathcal {Q}=({\mathcal D}\bar \otimes \ell ^\infty (\Gamma /\Sigma ))\rtimes \Gamma $ the semifinite von Neumann algebra arising from the diagonal action $\Gamma \curvearrowright {\mathcal D}\bar \otimes \ell ^\infty (\Gamma /\Sigma )$ .

Then there exists a $*$ -isomorphism $\theta : \langle {\mathcal M},e_{\mathcal N} \rangle \to \mathcal {Q}$ satisfying $\theta (x)=x$ , for all $x\in {\mathcal M}$ , and $\theta (e_{\mathcal N})=1\otimes \delta _{e\Sigma }\in {\mathcal D}\bar \otimes \ell ^\infty (\Gamma /\Sigma )$ .

Proof. We denote by $\tau $ the trace on ${\mathcal M}$ and by $\mathrm {Tr}$ the natural faithful normal semifinite trace on $\mathcal {Q}$ , so that the restriction of $(\mathcal {Q}, \mathrm {Tr})$ to ${\mathcal M}$ is $({\mathcal M},\tau )$ . By letting $f=1\otimes \delta _{e\Sigma }$ , note that the following properties hold:

1. (i) $\mathcal {Q}$ is the weak closure of the $*$ -subspace ${\mathcal M} f{\mathcal M}$ ,

2. (ii) $\mathrm {Tr}(f)=1$ and $\mathrm {Tr}(xf)=\tau (x)$ , for any $x\in {\mathcal M}$ ,

3. (iii) $f\mathcal {Q} f=f\mathcal N=\mathcal N f$ .

By using [SS10, Theorem 4.3.15], we derive from properties (i)–(iii) that there is a $*$ -isomorphism $\theta : \langle {\mathcal M},e_{\mathcal N} \rangle \to \mathcal {Q}$ with $\theta (x)=x$ , for all $x\in {\mathcal M}$ , and $\theta (e_{\mathcal N})=f$ .

Proposition 2.12. Let $\varepsilon :\Gamma \rightarrow \Lambda $ be an epimorphism between countable groups, and let $\Gamma \curvearrowright {\mathcal D}$ be a trace preserving action. Let ${\mathcal M} = {\mathcal D}\rtimes \Gamma $ and $\mathcal N= {\mathcal L}(\Lambda )$ and denote by $\Delta : {\mathcal M}\rightarrow {\mathcal M}\bar \otimes \mathcal N$ be the $\ast $ -embedding given by $\Delta (au_g)=au_g\otimes v_{\varepsilon (g)}$ for all $a\in {\mathcal D},g\in \Gamma $ . Here, we denoted by $\{u_g\}_{g\in \Gamma }$ the canonical unitaries that implement the action $\Gamma \curvearrowright {\mathcal D}$ and by $\{v_\lambda \}_{\lambda \in \Lambda }$ the canonical generating unitaries of ${\mathcal L}(\Lambda )$ .

If $\mathcal P\subseteq p {\mathcal M} p$ is a von Neumann subalgebra, such that $\Delta (\mathcal P)$ is amenable relative to ${\mathcal M}\otimes 1$ inside ${\mathcal M} \bar \otimes \mathcal N$ , then $\mathcal P$ is amenable relative to ${\mathcal D}\rtimes \ker (\varepsilon )$ inside ${\mathcal M}$ .

Proof. First, we claim that there is an injective $*$ -homomorphism $\theta : \langle {\mathcal M}, e_{{\mathcal D}\rtimes \ker (\varepsilon )} \rangle \to \langle {\mathcal M}\bar \otimes \mathcal N, e_{{\mathcal M}\otimes 1} \rangle $ which satisfies $\theta (x)=\Delta (x)$ , for any $x\in {\mathcal M}$ , and $\theta (e_{{\mathcal D}\rtimes \ker (\varepsilon )})=e_{{\mathcal M}\otimes 1}$ . To see that the above claim holds, we use Lemma 2.11 to obtain a $*$ -isomorphism $\theta _1: \langle {\mathcal M}, e_{{\mathcal D}\rtimes \ker (\varepsilon )} \rangle \to ({\mathcal D}\bar \otimes \ell ^{\infty }(\Gamma /\ker (\varepsilon )))\rtimes \Gamma $ which satisfies $\theta _1(au_g)=(a\otimes 1)u_g$ , for all $a\in {\mathcal D},g\in \Gamma $ and $\theta _1(e_{{\mathcal D}\rtimes \ker (\varepsilon )})=1\otimes \delta _{e \ker (\varepsilon )}$ . By applying once again Lemma 2.11, it follows that there is a $*$ -isomorphism $\theta _2: \langle {\mathcal M}\bar \otimes \mathcal N, e_{{\mathcal M}\otimes 1} \rangle \to ({\mathcal M}\bar \otimes \ell ^{\infty }(\Lambda ))\rtimes \Lambda $ which satisfies $\theta _2(au_g\otimes v_\lambda )=(au_g\otimes 1)v_\lambda $ , for all $a\in {\mathcal D},g\in \Gamma ,\lambda \in \Lambda $ and $\theta _2(e_{{\mathcal M}\otimes 1})=1\otimes \delta _e$ . Here, $({\mathcal M}\bar \otimes \ell ^{\infty }(\Lambda ))\rtimes \Lambda $ arises from the diagonal action of $\Lambda $ on ${\mathcal M}\bar \otimes \ell ^{\infty }(\Lambda )$ , where $\Lambda $ acts trivially on ${\mathcal M}$ and by left translations on $\ell ^\infty (\Lambda )$ .

Next, since the epimorphism $\varepsilon :\Gamma \to \Lambda $ naturally defines a group isomorphism between $\Gamma /\ker (\varepsilon )$ and $\Lambda $ , there is a $*$ -isomorphism between $\ell ^{\infty }(\Gamma /\ker (\varepsilon ))$ and $\ell ^\infty (\Lambda )$ which sends $\delta _{e \mathrm {ker}(\epsilon )}$ to $\delta _e$ . Therefore, we can define an injective $*$ -homomorphism $\varphi : ({\mathcal D}\bar \otimes \ell ^{\infty }(\Gamma /\ker (\varepsilon )))\rtimes \Gamma \to ({\mathcal M}\bar \otimes \ell ^{\infty }(\Lambda ))\rtimes \Lambda $ by letting $\varphi ((a\otimes f)u_g)=(au_g\otimes f)v_{\varepsilon (g)}$ , for all $a\in D,g\in \Gamma $ and $f\in \ell ^{\infty }(\Gamma /\ker (\varepsilon ))$ . Altogether, it implies that we can define an injective $*$ -homomorphism $\theta : \langle {\mathcal M}, e_{{\mathcal D}\rtimes \ker (\varepsilon )} \rangle \to \langle {\mathcal M}\bar \otimes \mathcal N, e_{{\mathcal M}\otimes 1} \rangle $ by letting $\theta =\theta _2^{-1}\circ \varphi \circ \theta _1$ . In this way, $\theta (au_g)=\theta _2^{-1} (\varphi ((a\otimes 1)u_g))=\theta _2^{-1}((au_g\otimes 1)v_{\varepsilon (g)})=\Delta (au_g)$ , for all $a\in {\mathcal D}$ and $g\in \Gamma $ . Note that we also have $\theta (e_{{\mathcal D}\rtimes \ker (\varepsilon )})=\theta _2^{-1}(\varphi (1\otimes \delta _{e \ker (\varepsilon )}))=\theta _2^{-1}(1\otimes \delta _e)= e_{{\mathcal M}\otimes 1}$ , hence, proving the above claim.

Finally, denote $\tilde p=\Delta (p)$ . The assumption implies that there exists a $\Delta (\mathcal P)$ -central positive linear functional $\Phi : \tilde p\langle {\mathcal M}\bar \otimes \mathcal N, e_{{\mathcal M}\otimes 1} \rangle \tilde p\to \mathbb C$ , such that the restriction of $\Phi $ to $\tilde p ({\mathcal M}\bar \otimes \mathcal N) \tilde p$ equals the trace on $\tilde p ({\mathcal M}\bar \otimes \mathcal N) \tilde p$ . Define now the positive linear functional $\Psi : p\langle {\mathcal M}, e_{{\mathcal D}\rtimes \ker (\varepsilon )} \rangle p \to \mathbb C$ by $\Psi (x)=(\Phi \circ \theta )(x)$ , for all $x\in p\langle {\mathcal M}, e_{{\mathcal D}\rtimes \ker (\varepsilon )} \rangle p$ , and note that the restriction of $\Psi $ to $p{\mathcal M} p$ equals the trace on $p{\mathcal M} p$ and $\Psi $ is $\mathcal P$ -central. This shows that $\mathcal P$ is amenable relative to ${\mathcal D}\rtimes \ker (\varepsilon )$ inside ${\mathcal M}$ .

2.7 Quasi-normalizers of von Neumann algebras

Given a group inclusion $\Sigma <\Gamma $ , the one-sided quasi-normalizer $\mathrm {QN}^{(1)}_\Gamma (\Sigma )$ is the semigroup of all $g\in \Gamma $ for which there exists a finite set $F\subset \Gamma $ , such that $\Sigma g\subset F \Sigma $ [FGS10, Section 5]; equivalently, $g\in \mathrm {QN}^{(1)}_\Gamma (\Sigma )$ if and only if $[\Sigma : g \Sigma g^{-1}\cap \Sigma ]<\infty $ . The quasi-normalizer $\mathrm { QN}_\Gamma (\Sigma )$ is the group of all $g\in \Gamma $ for which exists a finite set $F\subset \Gamma $ , such that $\Sigma g\subset F\Sigma $ and $g \Sigma \subset \Sigma F$ .

Given an inclusion $\mathcal P \subseteq {\mathcal M}$ of finite von Neumann algebra, we define the quasi-normalizer $\mathscr {QN}_{{\mathcal M}}(\mathcal P)$ as the set of all elements $x\in {\mathcal M}$ for which there exist $x_1,...,x_n\in {\mathcal M}$ , such that $\mathcal P x\subseteq \sum x_i \mathcal P$ and $x\mathcal P \subseteq \sum \mathcal P x_i$ (see [Po99, Definition 4.8]). Also, the one-sided quasi-normalizer $\mathscr {QN}^{(1)}_{{\mathcal M}}(\mathcal P)$ is defined as the set of all elements $x\in {\mathcal M}$ for which there exist $x_1,...,x_n\in {\mathcal M}$ such that $\mathcal P x\subseteq \sum x_i \mathcal P$ [FGS10]. We continue by recalling from [Po03, FGS10] some basic properties of (one-sided) quasi-normalizing algebras.

Lemma 2.13 [Po03, FGS10].

Let $\mathcal P\subset {\mathcal M}$ be tracial von Neumann algebras. For any projection $p\in \mathcal P$ , the following hold:

1. 1. $W^*({\mathscr {QN}^{(1)}_{p{\mathcal M} p}}(p\mathcal P p))=pW^*(\mathrm {\mathscr {QN}^{(1)}_{{\mathcal M}}}(\mathcal P))p$ .

2. 2. $W^*({ \mathscr {QN}_{p{\mathcal M} p}}(p\mathcal P p))=pW^*(\mathrm {\mathscr {QN}_{M}}(\mathcal P))p$ .

We continue by recording the following remark, which can be deduced directly from the definition.

Remark 2.14. Let $\mathcal P\subset {\mathcal M}$ be tracial von Neumann algebras. For any projection ${p\in \mathcal P'\cap {\mathcal M}}$ , we have $W^*(\mathrm {\mathscr {QN}_{p'{\mathcal M} p'}}(\mathcal P p'))=p'W^*(\mathrm { \mathscr {QN}_{{\mathcal M}}}(\mathcal P))p'$ .

Finally, we record the following result, which provides a relation between the group theoretical quasi-normalizer and the von Neumann algebraic one.

Lemma 2.15 (Corollary 5.2 in [FGS10]).

Let $\Sigma <\Gamma $ be countable groups. Then the following hold:

1. 1. $W^*(\mathscr {QN}^{(1)}_{{\mathcal L}(\Gamma )}(\mathcal L(\Sigma )))=\mathcal L(K)$ , where $K<\Gamma $ is the subgroup generated by $\mathrm {QN}^{(1)}_\Gamma (\Sigma )$ . In particular, if $\mathrm { QN}^{(1)}_\Gamma (\Sigma )=\Sigma $ , then $\mathscr {QN}^{(1)}_{{\mathcal L}(\Gamma )}({\mathcal L}(\Sigma ))={\mathcal L}(\Sigma )$ .

2. 2. $W^*(\mathscr {QN}_{{\mathcal L}(\Gamma )}({\mathcal L}(\Sigma )))=\mathcal L(\mathrm {QN}_\Gamma (\Sigma ))$ .

For further use, we briefly recall some results from [CDD22] that provide control of quasi-normalizers of certain von Neumann subalgebras in von Neumann algebras of general graph product groups.

Theorem 2.16 [CDD22, Theorem 2.7].

Let $\Gamma =\mathscr {G} \{\Gamma _v\}$ be any graph product of groups, and let $\mathscr {S},\mathscr {T}\subseteq \mathscr {G}$ be any subgraphs. Denote by ${\mathcal M}= {\mathcal L}(\Gamma )$ , and assume there exist $x,x_1,x_2,...,x_n \in {\mathcal M}$ , such that ${\mathcal L}(\Gamma _{\mathscr {S}})x\subseteq \sum ^n_{k=1} x_k \mathcal L(\Gamma _{\mathscr {T}})$ . Then $\mathscr {S}\subseteq \mathscr {T}$ and $x\in \mathcal L(\Gamma _{\mathscr {T} \cup \mathrm {lk}(\mathscr {S}) })$ .

Corollary 2.17. Let $\Gamma =\mathscr {G} \{\Gamma _v\}$ be any graph product of groups, and let $\mathscr {C}\in \mathrm {cliq}( \mathscr {G})$ be a clique with at least two vertices. Fix a vertex $v\in \mathscr {C}$ , such that $\mathrm {lk}(\mathscr {C} \setminus \{v\})=\{v\}$ . Denote by ${\mathcal M}= {\mathcal L}(\Gamma )$ , and assume there exist $x,x_1,x_2,...,x_n \in {\mathcal M}$ , such that $\mathcal L(\Gamma _{\mathscr {C} \setminus \{v\}})x\subseteq \sum ^n_{k=1} x_k {\mathcal L}(\Gamma _{\mathscr {C}})$ . Then $x\in {\mathcal L}(\Gamma _{\mathscr {C} })$ .

Proof. Follows applying Theorem 2.16 for $\mathscr {S} =\mathscr {C} \setminus \{v\} $ and $\mathscr {T} =\mathscr {C}$ .

Lemma 2.18 [CDD22, Lemma 2.9].

Let $\Gamma =\mathscr {G} \{\Gamma _v\}$ be a graph product of groups, and let $\mathscr {C}\in \mathrm {cliq}(\mathscr {G})$ be a clique. Let $\mathcal P\subset p{\mathcal L}(\Gamma _{\mathscr {C}})p$ be a von Neumann subalgebra, such that $\mathcal P\nprec _{{\mathcal L}(\Gamma _{\mathscr {C}})} L(\Gamma _{\mathscr {C} \setminus \{v\}})$ , for any $v\in \mathscr {C}$ . If $x\in {\mathcal L}(\Gamma )$ satisfies $x\mathcal P\subset \sum _{i=1}^n {\mathcal L}(\Gamma _{\mathscr {C}})x_i$ for some $x_1,\dots ,x_n\in {\mathcal L}(\Gamma )$ , then $xp\in {\mathcal L}(\Gamma _{\mathscr {C}})$ .

3 A version of Popa-Vaes dichotomy for normalizers of subalgebras in crossed-product von Neumann algebras

In [PV12, Theorem 1.4], Popa and Vaes established a remarkably deep dichotomy for normalizers of von Neumann subalgebras in crossed-product von Neumann algebras, ${\mathcal A\rtimes \Gamma} $ , arising from trace preserving actions $\Gamma \curvearrowright {\mathcal D}$ of biexact weakly amenable groups $\Gamma $ on tracial von Neumann algebras ${\mathcal D}$ . For further use, we present, next, a version of this result for actions of groups which surject onto biexact, weakly amenable groups.

Theorem 3.1. Let $\Sigma $ be a biexact weakly amenable group, and let $\Gamma $ be a group which admits an epimorphism $\varepsilon : \Gamma \rightarrow \Sigma $ . Let ${\mathcal D}$ be a tracial von Neumann algebra, and let $\Gamma \curvearrowright ^{\sigma } {\mathcal D}$ be a trace-preserving action. Denote by ${\mathcal M} ={\mathcal D} \rtimes _{\sigma } \Gamma $ the corresponding crossed-product von Neumann algebra, and let $0\neq p\in {\mathcal M}$ be a projection. Then, for any von Neumann subalgebra $\mathcal P \subset p{\mathcal M} p$ that is amenable relative to ${\mathcal D}$ inside ${\mathcal M}$ , one of the following must hold:

1. 1. $\mathcal P \prec _{\mathcal M} {\mathcal D}\rtimes \ker (\varepsilon )$ , or

2. 2. $\mathscr N_{p{\mathcal M} p}(\mathcal P)"$ is amenable relative to ${\mathcal D}\rtimes \ker (\varepsilon ) $ inside ${\mathcal M}$ .

Proof. Denote $\mathcal {Q}=\mathscr N_{p{\mathcal M} p}(\mathcal P)"$ . Following [CIK13, Section 3], consider the $\ast $ -embedding $\Delta : {\mathcal M} \rightarrow {\mathcal M} \bar \otimes {\mathcal L}(\Sigma )$ given by $\Delta (au_g)=(au_g) \otimes v_{\varepsilon (g)}$ for all $a\in {\mathcal D}$ and $g\in \Gamma $ , where $(v_g)_{g\in \Gamma }$ are the canonical group unitaries in ${\mathcal L}(\Sigma )$ . Then, using [PV12, Theorem 1.4], one of the following must hold:

1. a) $\Delta (\mathcal P)\prec _{{\mathcal M}\bar \otimes {\mathcal L}(\Sigma )} {\mathcal M}\otimes 1$ , or

2. b) $\Delta (\mathcal {Q})$ is amenable relative to ${\mathcal M} \otimes 1$ inside ${\mathcal M}\bar \otimes {\mathcal L}(\Sigma )$ .

Using [CIK13, Proposition 3.4], one can see that case a) implies 1. Moreover, using Proposition 2.12, case b) yields 2.

4 Solidity results for von Neumann algebras of wreath-like product groups

We start this section by recording the following result, which follows from Popa and Vaes’s structure theorem [PV12, Theorem 1.4] for normalizers inside crossed products of hyperbolic groups, as in [CIK13].

Proposition 4.1. Let $\Gamma \in \mathcal W\mathcal R (A, B \curvearrowright I)$ , where A is Abelian and B is a subgroup of a hyperbolic group. Let $\Gamma \curvearrowright {\mathcal D}$ be a trace preserving action, and denote ${\mathcal M}={\mathcal D}\rtimes \Gamma .$ Let $\mathcal A,\mathcal B\subset p {\mathcal M} p$ be commuting von Neumann subalgebras.

Then, either $\mathcal A\prec _{{\mathcal M}} {\mathcal D}\rtimes A^{I}$ or $\mathcal B$ is amenable relative to ${\mathcal D}$ inside ${\mathcal M}$ .

Proof. Let $\varepsilon :\Gamma \to B$ be the quotient homomorphism given by $\Gamma \in \mathcal W\mathcal R (A, B \curvearrowright I)$ , and let $\{u_g\}_{g\in \Gamma }$ and $\{v_h\}_{h\in B}$ be the canonical generating unitaries of ${\mathcal L}(\Gamma )$ and ${\mathcal L}(B)$ , respectively. Denote by $\Delta :{\mathcal M}\to {\mathcal M}\bar \otimes \mathcal L(B)$ the $*$ -homomorphism given by $\Delta (au_g)=au_g\otimes v_{\varepsilon (g)}$ , for all $a\in {\mathcal D},g\in \Gamma $ . From [KV15, Lemma 5.2], we derive that either $\Delta (\mathcal A)\prec _{{\mathcal M}\bar \otimes {\mathcal L}(B)} {\mathcal M}\otimes 1$ or $\Delta (\mathcal B)$ is amenable relative to ${\mathcal M}\bar \otimes 1$ . The first possibility implies by [CIK13, Proposition 3.4] that $\mathcal A\prec _{{\mathcal M}} {\mathcal D}\rtimes A^{I}$ , while the second one implies by Proposition 2.12 that $\mathcal B$ is amenable relative to ${\mathcal D}\rtimes A^{I}$ inside ${\mathcal M}$ . Since A is amenable, the conclusion follows from [OP07, Proposition 2.4].

We recall that a II $_1$ factor ${\mathcal M}$ is called solid if for any projection $p\in {\mathcal M}$ and any diffuse von Neumann subalgebra $\mathcal A \subset p {\mathcal M} p$ the relative commutant $\mathcal A'\cap p{\mathcal M} p$ is amenable. We note that if ${\mathcal M}$ is solid, then any amplification ${\mathcal M}^t$ , where $t>0$ , is also solid.

Theorem 4.2. Let $\Gamma \in \mathcal W\mathcal R (A, B \curvearrowright I)$ where A is Abelian, B is a subgroup of a hyperbolic group and the action $B\curvearrowright I$ has amenable stabilizers. Then ${\mathcal L}(\Gamma )$ is solid.

Proof. Let $\mathcal A \subset p{\mathcal L}(\Gamma )p$ be a diffuse von Neumann subalgebra, and assume by contradiction that $\mathcal B:=\mathcal A'\cap p{\mathcal L}(\Gamma )p$ is nonamenable. Thus, there exists a nonzero projection ${z\in \mathcal Z(\mathcal B)}$ , such that $ q \mathcal B q$ is nonamenable for every projection $q \in \mathcal B z$ . By applying Proposition 4.1, we deduce that $\mathcal A z\prec ^s_{{\mathcal L}(\Gamma )} {\mathcal L}(A^{I})$ , and hence, by using [CIOS21, Corollary 4.7] (for $m=1$ ) and also [CIOS21, Lemma 4.11], we get that $\mathcal B z$ is amenable, contradiction.

Remark 4.3. Notice that this result generalizes some of the results obtained in [CIOS21, Section 9.2].

Proposition 4.4. For any $1\leq i\leq n$ , we consider $\Gamma _i\in \mathcal W\mathcal R (A_i, B_i \curvearrowright I_i)$ where $A_i$ is Abelian, $B_i$ is a subgroup of a hyperbolic group and denote $\Gamma = \Gamma _1\times \cdots \times \Gamma _n$ . Let $\Lambda $ be a countable group, and denote ${\mathcal M}={\mathcal L}(\Lambda )$ . Assume that ${\mathcal L}(\Gamma )\subset {\mathcal M}$ is a von Neumann subalgebra.

Let $\mathcal P_1,\dots ,\mathcal P_{m}\subset p{\mathcal M} p$ be commuting nonamenable factors and $\Lambda _0<\Lambda $ a subgroup, such that $\mathcal P_1\vee \dots \vee \mathcal P_{m}\prec _{{\mathcal M}} {\mathcal L}(\Lambda _0)$ and $\mathcal L(\Lambda _0)\prec _{{\mathcal M}} \mathcal L(\Gamma _1\times \dots \times \Gamma _n)$ . Then, $m\leq n$ .

Throughout the proof, we use the following notation. If $\mathcal P\subset p{\mathcal M} p, \mathcal {Q}\subset q {\mathcal M} q$ are subalgebras, such that $\mathcal P\prec _{{\mathcal M}} \mathcal {Q} q'$ , for all projections $0\neq q'\in \mathcal {Q}'\cap q{\mathcal M} q$ , we write $\mathcal P\prec _{{\mathcal M}}^{s'} \mathcal {Q}$ .

Proof. Denote $\Gamma =\Gamma _1\times \dots \times \Gamma _n$ and $\mathcal P=\mathcal P_1\vee \dots \vee \mathcal P_{m}$ . Following the augmentation technique from [CD-AD20, Section 3], we consider a Bernoulli action with Abelian base $\Lambda \curvearrowright {\mathcal D}^\Lambda $ , and let $\tilde {\mathcal M}={\mathcal D}^\Lambda \rtimes \Lambda $ . Let $\Delta :\tilde {{\mathcal M}}\to \tilde {{\mathcal M}}\bar \otimes {\mathcal M}$ be the $*$ -homomorphism given by $\Delta (dv_h)=dv_h\otimes v_h$ , for all $d\in {\mathcal D}^\Lambda , h\in \Lambda $ . The assumption implies that $\mathcal P\prec ^{s'}_{\tilde {\mathcal M}} {\mathcal D}^\Lambda \rtimes \Lambda _0$ , and hence, by applying [Dr19b, Lemma 2.3], we get that $\Delta (\mathcal P)\prec _{\tilde {\mathcal M}\bar \otimes {\mathcal M}}^{s'} \tilde {{\mathcal M}} \bar \otimes {\mathcal L}(\Lambda _0)$ . Next, from [Dr19b, Lemma 2.4(2)], we derive that $\Delta (\mathcal P)\prec _{\tilde {\mathcal M}\bar \otimes {\mathcal M}} \tilde {{\mathcal M}} \bar \otimes {\mathcal L}(\Gamma )$ .

Thus, there exist projections $r\in \mathcal P, q\in \tilde {{\mathcal M}} \bar \otimes {\mathcal L}(\Gamma )$ , a nonzero partial isometry ${v\in q(\tilde {{\mathcal M}}\bar \otimes {\mathcal M})\Delta (r)}$ and a $*$ -homomorphism $\theta : \Delta (r\mathcal P r)\to q (\tilde {{\mathcal M}} \bar \otimes {\mathcal L}(\Gamma )) q$ , such that $\theta (x)v=vx$ , for any $x\in \Delta (rPr)$ , and the support projection of $E_{\tilde {{\mathcal M}} \bar \otimes {\mathcal L}(\Gamma )}(vv^*)$ equals q. Using [CdSS17, Lemma 4.5], we can assume that $r\in \mathcal P_1$ . Denote $\mathcal N=\tilde {{\mathcal M}} \bar \otimes {\mathcal L}(\Gamma )$ and $\mathcal {Q}_i=\Delta (r\mathcal P_i r)$ for any $1\leq i\leq m$ . From [IPV10, Lemma 10.2], it follows that $\Delta (\mathcal P_i)$ is strongly nonamenable relative to $\tilde {{\mathcal M}}\otimes 1$ inside $\tilde {{\mathcal M}}\bar \otimes {\mathcal M}$ , for any $1\leq i\leq m$ . Therefore, $\mathcal {Q}_i$ is nonamenable relative to $\tilde {{\mathcal M}}\otimes 1$ inside $\mathcal N$ for any $1\leq i\leq m$ . In particular, we derive from [PV11, Proposition 2.7] that there exists $1\leq j\leq n$ , such that $\mathcal {Q}_m$ is nonamenable relative to $\tilde {{\mathcal M}}\otimes {\mathcal L}(\Gamma _{\widehat {j}})$ inside $\mathcal N$ . Without loss of generality, we may assume that $j=n$ . By applying Proposition 4.1, it follows that $\bigvee _{i=1}^{m-1} Q_i\prec _{\mathcal N} \tilde {{\mathcal M}}\otimes \mathcal L(\Gamma _{\widehat {n}}\times A_{n}),$ where $A_n<\Gamma _{n}$ is an amenable subgroup. Thus, $ \Delta (\bigvee _{i=1}^{m-1} \mathcal P_i)\prec _{\tilde {{\mathcal M}}\bar \otimes {\mathcal M}} \tilde {{\mathcal M}}\bar \otimes {\mathcal L}(\Gamma _{\widehat {n}}\times A_{n}). $

Assume, by contradiction, that $m>n$ . By repeating the previous argument finitely many times, we obtain that there exists an amenable subgroup $A<\Gamma $ with the property that $\Delta (\bigvee _{i=1}^{m-n} \mathcal P_i)\prec _{\tilde {{\mathcal M}}\bar \otimes {\mathcal M}}\tilde {{\mathcal M}}\bar \otimes {\mathcal L}(A)$ . By applying [IPV10, Lemma 10.2], it follows that $\mathcal P_1$ is amenable, contradiction.

Proposition 4.5. For any $1\leq i\leq n$ , we consider $\Gamma _i\in \mathcal W\mathcal R (A_i, B_i \curvearrowright I_i)$ where $A_i$ is Abelian, $B_i$ is a subgroup of a hyperbolic group and denote $\Gamma = \Gamma _1\times \cdots \times \Gamma _n$ . Let $\mathcal P_0,\mathcal P_1,..., \mathcal P_n\subseteq p {\mathcal L}(\Gamma ) p$ be commuting von Neumann subalgebras, such that $\mathcal P_n$ has no amenable direct summand and $\mathcal P_i$ is a nonamenable factor for any $1\leq i\leq n-1$ .

Then $\mathcal P_0$ is completely atomic.

Proof. Throughout the proof, we denote by $\Gamma _S=\times _{i\in S}\Gamma _i$ the subproduct supported on a subset $S\subset \{1,\dots , n\}$ . We also denote by $\widehat S$ the complement of S inside $\{1,\dots , n\}$ . Denote ${\mathcal M}={\mathcal L}(\Gamma )$ . Since $\mathcal P_n$ has no amenable direct summand, it follows that $\mathcal P_n z$ is nonamenable for any nonzero projection $z\in (\bigvee _{i=0}^n \mathcal P_i)'\cap pMp$ . By [PV11, Proposition 2.7], we get that there exists $1\leq j\leq n$ , such that $\mathcal P_n z$ is nonamenable relative to ${\mathcal L}(\Gamma _{\widehat j})$ inside ${\mathcal M}$ . Without loss of generalization, assume $j=n$ . Using Proposition 4.1, we get that $(\bigvee _{i=0}^{n-1} \mathcal P_i)z\prec _{{\mathcal M}} {\mathcal L}(\Gamma _{\widehat n})\bar \otimes {\mathcal L}(A_n^{(I_n)})$ . Thus, there exist projections $r\in \bigvee _{i=0}^{n-1} \mathcal P_i, q\in {\mathcal L}(\Gamma _{\widehat n})\bar \otimes \mathcal L(A_n^{(I_n)})$ with $rz\neq 0$ , a nonzero partial isometry $v\in q{\mathcal M} r$ and a $*$ -homomorphism $\theta : r(\bigvee _{i=0}^{n-1} \mathcal P_i)rz \to q({\mathcal L}(\Gamma _{\widehat n})\bar \otimes \mathcal L(A_n^{(I_n)}))q$ , such that $\theta (x)v=vx$ and the support projection of $E_{\mathcal L(\Gamma _{\widehat n})\bar \otimes \mathcal L(A_n^{(I_n)})}(vv^*)$ equals q. Since $\mathcal P_{n-1}$ is a II $_1$ factor, by using [CdSS17, Lemma 4.5], we can assume that $r\in \mathcal P_{n-1}$ . Denote $\mathcal N=q({\mathcal L}(\Gamma _{\widehat n})\bar \otimes {\mathcal L}(A_n^{(I_n)}))q$ and $\mathcal {Q}_i=\theta (r\mathcal P_irz)$ for any $0\leq i\leq n-1$ . Since $\mathcal {Q}_{n-1}$ is nonamenable, we use [PV11, Proposition 2.7] and derive that exists $1\leq k\leq n-1$ , such that $\mathcal {Q}_{n-1}$ is nonamenable relative to $\mathcal L( \Gamma _{\widehat {k,n}})\bar \otimes {\mathcal L}(A_k^{(I_k)}\times A_n^{(I_n)})$ inside $\mathcal N$ . Without loss of generalization, we assume $k=n-1$ . By applying Proposition 4.1, we derive that $ \bigvee _{i=0}^{n-2} \mathcal {Q}_i\prec _{\mathcal N} {\mathcal L}( \Gamma _{\widehat {n-1,n}})\bar \otimes {\mathcal L}(A_{n-1}^{(I_{n-1})}\times A_n^{(I_n)}). $ Using that $\mathcal {Q}_i v=v r\mathcal P_i rz$ , we derive that $(\bigvee _{i=0}^{n-2} \mathcal P_i)z\prec _{{\mathcal M}} {\mathcal L}(\Gamma _{\widehat {n-1,n}})\bar \otimes {\mathcal L}(A_{n-1}^{(I_{n-1})}\times A_n^{(I_n)})$ . By using the previous argument finitely many times, we obtain $\mathcal P_0z\prec _{{\mathcal M}}{\mathcal L}(\times _{i=1}^n A_i^{(I_i)})$ .

Assume, by contradiction, that $\mathcal P_0z\nprec _{{\mathcal M}} \mathbb C1$ . The previous paragraph implies that there exists a nonempty subset $F\subset \{1,\dots ,n\}$ , such that

(4.1) $$ \begin{align} \mathcal P_0z\prec_{{\mathcal M}}{\mathcal L}(\times_{i\in F} A_i^{(I_i)}) \text{ and } \mathcal P_0z\nprec_{{\mathcal M}}{\mathcal L}(\times_{i\in F\setminus\{j\}} A_i^{(I_i)}) \text{ for any }j\in F. \end{align} $$

Therefore, there exist projections $p_0\in P_0, s\in {\mathcal L}(\times _{i\in F} A_i^{(I_i)})$ with $p_0z\neq 0$ , a nonzero partial isometry $w\in s{\mathcal M} p_0$ , a $*$ -homomorphism $\Psi : p_0P_0p_0z\to s{\mathcal L}(\times _{i\in F} A_i^{(I_i)})s$ , such that $\Psi (x)w=wx$ , for any $x\in p_0P_0p_0z$ , and the support projection of $E_{\mathcal L(\times _{i\in F} A_i^{(I_i)})}(ww^*)$ equals s. By letting $T=\Psi (p_0P_0z)$ and $A=\times _{i\in F} A_i^{(I_i)}$ , one can check that (4.1) implies that

$$ \begin{align*}T\subset s{\mathcal L}(A)s \text{ and } T\nprec_{{\mathcal L}(A)}{\mathcal L}(\times_{i\in F\setminus\{j\}} A_i^{(I_i)}) \text{ for any }j\in F. \end{align*} $$

By applying [CIOS21, Corollary 4.7 and Lemma 4.11], we have $T'\cap s {\mathcal L}(\Gamma _{F}) s$ is amenable. Note that the $*$ -isomorphism $\mathrm {Ad}(w): w^*w {\mathcal M} w^*w\to ww^* {\mathcal M} ww^*$ sends $w^*w (p_0P_0p_0z'\cap p_0z{\mathcal M} p_0z) w^*w$ onto $ww^* (T'\cap s{\mathcal M} s) ww^*$ . Hence,

$$ \begin{align*} w ( p_0P_0p_0z'\cap p_0z{\mathcal M} p_0z ) w^*=ww^* ((T'\cap s {\mathcal L}(\Gamma_{F}) s)\bar\otimes {\mathcal L}(\Gamma_{\widehat F})s ) ww^*. \end{align*} $$

Since $\bigvee _{i=1}^n \mathcal P_i\subset \mathcal P_0'\cap p{\mathcal M} p$ , it follows that $\bigvee _{i=1}^n \mathcal P_i\prec _{{\mathcal M}} (T'\cap s {\mathcal L}(\Gamma _{F}) s)\bar \otimes \mathcal L(\Gamma _{\widehat F})s $ . Since $\widehat F$ has at most $n-1$ elements and $T'\cap s \mathcal L(\Gamma _{F}) s$ is amenable, we can apply the previous arguments finitely many times and derive that there exists an amenable subalgebra $S\subset {\mathcal M} $ , such that $\mathcal P_n\prec _{{\mathcal M}} S$ . This shows that $\mathcal P_n$ has an amenable direct summand, contradiction.

Finally, we conclude that $\mathcal P_0z\prec _{{\mathcal M}} \mathbb C1$ , for all nonzero projections $z\in (\bigvee _{i=0}^n \mathcal P_i)'\cap p{\mathcal M} p$ . Using [DHI16, Lemma 2.4], we conclude that $\mathcal P_0\prec ^s_{{\mathcal M}} \mathbb C1$ , which implies that $\mathcal P_0$ is completely atomic.

5 Product rigidity results for von Neumann algebras of wreath-like product groups

The goal of this section is to prove Theorem A and consequently derive Corollary B. We first prove the following proposition.

Proposition 5.1 [CU18, CdSS15].

Let $\Gamma $ be an icc group, denote ${\mathcal M}=L(\Gamma )$ and assume that ${\mathcal M}=\mathcal A \bar \otimes \mathcal B$ where $\mathcal B$ is solid. Let $ \Omega <\Gamma $ be a subgroup with nonamenable centralizer $C_\Lambda (\Omega )$ , such that $\mathcal A\prec {\mathcal L}(\Omega )$ .

Then there exist projections $e\in {\mathcal L}(\Omega )$ and $r \in {\mathcal L}(\Omega )'\cap {\mathcal M}$ with $e r\neq 0$ , such that $ e(\mathcal L(\Omega )e r \vee r(\mathcal L(\Omega )'\cap {\mathcal M})r e\subseteq er{\mathcal M} er $ is a finite index inclusion of II $_1$ factors.

Proof. Let $\mathcal {Q}={\mathcal L}(\Omega )$ as $\mathcal A\prec \mathcal {Q}$ , then using [CKP14, Proposition 2.4] and its proof, one can find nonzero projections $a \in \mathcal A$ , $q \in \mathcal {Q}$ , a partial isometry $v\in q {\mathcal M} a$ , a subalgebra ${\mathcal D} \subseteq q\mathcal {Q} q$ and a $\ast $ -isomorphism $\phi : a\mathcal A a \rightarrow {\mathcal D} $ , such that

1. 1. ${\mathcal D} \vee ({\mathcal D}'\cap q\mathcal {Q} q) \subseteq q\mathcal {Q} q$ has finite index, and

2. 2. $\phi (x)v = vx$ for all $x \in a\mathcal A a$ .

The intertwining relation 2. implies that $vv^* \in {\mathcal D}' \cap q{\mathcal M} q$ and $v^*v \in (a\mathcal A a)' \cap a{\mathcal M} a = a \otimes \mathcal B.$ Hence, there is a projection $b \in \mathcal B$ satisfying $v ^*v = a \otimes b$ . Choosing $u \in \mathscr {U}({\mathcal M} )$ , such that $v = u(a \otimes b) $ then relation 2. entails that

(5.1) $$ \begin{align}{\mathcal D} vv^* = v\mathcal A v^* = u(a\mathcal A a \otimes b)u^*.\end{align} $$

Using this relation and passing to the relative commutants, we obtain $vv^* ({\mathcal D}' \cap q{\mathcal M} q)vv^* = u(a \otimes b\mathcal B b)u ^*$ . Thus, one can find scalars $s_1,s_2> 0$ satisfying

(5.2) $$ \begin{align}({\mathcal D}' \cap q{\mathcal M} q)z = u(a \otimes b\mathcal B b) ^{s_1} u ^* \cong \mathcal B ^{s_2}.\end{align} $$

Here, z denotes the central support projection of $vv^*$ in ${\mathcal D}'\cap q{\mathcal M} q$ . Now notice that

(5.3) $$ \begin{align} {\mathcal D}' \cap q {\mathcal M} q \supseteq (q\mathcal{Q} q)' \cap q{\mathcal M} q = (\mathcal{Q}' \cap {\mathcal M} )q \supseteq {\mathcal L}(C_\Lambda(\Omega))q,\end{align} $$

where ${\mathcal L}(C_\Lambda (\Omega ))$ has no amenable direct summand since $C_\Lambda (\Omega )$ is a nonamenable group. Moreover, we also have ${\mathcal D}' \cap q{\mathcal M} q \supseteq {\mathcal D}' \cap q \mathcal {Q} q$ . Thus, $(\mathcal {Q} ' \cap {\mathcal M} )z$ and $({\mathcal D}' \cap q\mathcal {Q} q)z$ are commuting subalgebras of $({\mathcal D}' \cap q {\mathcal M} q)z$ where $(\mathcal {Q}' \cap {\mathcal M} )z$ has no amenable direct summand. Since $\mathcal B^{s_2}$ is solid, then $({\mathcal D}' \cap q\mathcal {Q} q)z$ must be purely atomic. Thus, cutting by a central projection $0\neq r \in {\mathcal D}' \cap q\mathcal {Q} q$ , we may assume that ${\mathcal D} \subseteq q\mathcal {Q} q$ is a finite index inclusion of algebras. Since ${\mathcal D}$ is a factor, shrinking r if necessary, we can actually assume that ${\mathcal D} \subseteq q\mathcal {Q} q$ is an irreducible inclusion of finite index II $_1$ factors. Moreover, one can check that if one replaces v by the partial isometry of the polar decomposition of $r v \neq 0$ , then all relations (5.1), (5.2) and (5.3) are still satisfied. In addition, we can assume without any loss of generality that the support projection satisfies $s(E_{\mathcal {Q}}(vv^*)) = q$ .

Using [Jo81, Lemma 3.1], one can find a projection $e\in q\mathcal {Q} q$ and a subfactor $\mathcal R \subseteq {\mathcal D} \subseteq q \mathcal {Q} q$ , such that $e \mathcal {Q} e =\mathcal R e$ and the index $[{\mathcal D}:\mathcal R]=[q \mathcal {Q} q: {\mathcal D}]$ . Now, notice the restriction $\phi ^{-1}: \mathcal R \rightarrow a\mathcal A a$ is an injective $\ast $ -homomorphism, such that $\mathcal T:= \phi ^{-1}(\mathcal R)\subseteq a\mathcal A a$ is finite index and

(5.4) $$ \begin{align} \phi^{-1}(y)v^*=v^*y \text{ for all } y\in \mathcal R. \end{align} $$

Let $\phi ': \mathcal R e \rightarrow \mathcal R$ be the $\ast $ -homomorphism given by $\phi '(y e)=y$ , for any $y\in \mathcal R$ . Since e has full support in $\langle {\mathcal D}, e\rangle =q\mathcal {Q} q$ , we have $ev\neq 0$ . Letting $w_0$ be a partial isometry, such that $w_0 |v^*e |=v^*e$ , the (5.4) gives that $\theta = \phi ^{-1} \circ \phi ': e \mathcal {Q} e \rightarrow a\mathcal A a$ is an injective $\ast $ -isomorphism satisfying $\theta (e \mathcal {Q} e)= \mathcal T$ and

(5.5) $$ \begin{align} \theta(y) w_0^*=w_0^* y \text{ for all } y\in e \mathcal{Q} e. \end{align} $$

Notice that $w_0^*w_0\in \mathcal T'\cap (a \mathcal A a \bar \otimes \mathcal B)=(\mathcal T'\cap a \mathcal A a) \bar \otimes \mathcal B $ . Since we have $\mathcal T'\cap (a\mathcal A a\bar \otimes \mathcal B )\supseteq \mathscr Z(\mathcal T'\cap a \mathcal A a) \bar \otimes \mathcal B$ and these von Neumann algebras have the same center then proceeding as in the proof of [OP03, Proposition 12], one can see that $w_0^*w_0$ is equivalent in ${\mathcal T'\cap (a \mathcal A a \bar \otimes \mathcal B)}$ with projection in $\mathscr Z(\mathcal T'\cap a \mathcal A a) \bar \otimes \mathcal B$ . Thus, one can assume without any loss of generality that $w_0^*w_0\in \mathscr Z(\mathcal T'\cap a \mathcal A a) \bar \otimes \mathcal B$ . As $[a\mathcal A a:\mathcal T]<\infty $ , then $\mathcal T'\cap a\mathcal A a$ is finite-dimensional. Thus, replacing $w_0$ by $w=w_0r_0$ for a minimal projection $r_0\in \mathscr Z(\mathcal T'\cap a \mathcal A a)$ with $r_0 w_0^*|v^*e|\neq 0$ , we see all previous relations are satisfied, including (5.5). Moreover, we can assume that $w^*w= z_1\otimes z_2$ for some $z_1\in \mathscr Z(\mathcal T'\cap a \mathcal A a)$ and $z_2\in B$ . Hence, (5.5) implies

(5.6) $$ \begin{align} w^* \mathcal{Q} w= \theta(e \mathcal{Q} e) w^*w= \mathcal T z_1 \otimes z_2. \end{align} $$

Since $\mathcal T \subseteq a \mathcal A a$ is a finite index inclusion of II $_1$ factors, using the local index formula [Jo81, Lemma 2.2.1], we have that $\mathcal T z_1\subseteq z_1 \mathcal A z_1$ is a finite index inclusion of II $_1$ factors as well. In addition, we have

(5.7) $$ \begin{align} (w^* \mathcal{Q} w)'\cap (z_1\otimes z_2) {\mathcal M} (z_1\otimes z_2)= ((\mathcal T z_1)' \cap z_1 \mathcal A z_1) \bar\otimes z_2 \mathcal B z_2. \end{align} $$

Altogether, the previous relations (5.6) and (5.7) imply that

(5.8) $$ \begin{align} \begin{aligned} \mathcal T z_1 \bar\otimes z_2 \mathcal B z_2& \subset \mathcal T z_1 \vee (\mathcal T z_1'\cap z_1 \mathcal A a)\bar\otimes z_2 \mathcal B z_2\\ &= w^* \mathcal{Q} w\vee (w^* \mathcal{Q} w)'\cap (z_1\otimes z_2) {\mathcal M} (z_1\otimes z_2)\\ &\subseteq z_1 \mathcal A z_1\bar\otimes z_2 \mathcal B z_2 = (z_1\otimes z_2) {\mathcal M} (z_1\otimes z_2). \end{aligned} \end{align} $$

Since $\mathcal T z_1\subseteq z_1 \mathcal A z_1$ is a finite index inclusion of II $_1$ factors, then so is $\mathcal T z_1 \bar \otimes z_2 \mathcal B z_2\subseteq z_1 \mathcal A z_1\bar \otimes z_2 \mathcal B z_2$ . Let $f= ww^*$ , and note that $f=re$ for some projection $r\in \mathcal {Q}'\cap {\mathcal M}$ . Now notice (5.8) implies that

(5.9) $$ \begin{align} e\mathcal{Q} e r \vee r (\mathcal{Q} '\cap {\mathcal M})r e = e\mathcal{Q} e r \vee ((e\mathcal{Q} er) '\cap er{\mathcal M} er)\subseteq f {\mathcal M} f \end{align} $$

is a finite index inclusion of von Neumann algebras. Since ${\mathcal M}$ is a factor, the finite index condition implies that the center $\mathscr Z(r(\mathcal {Q}'\cap {\mathcal M} )r)$ is completely atomic (see [CdSS17, Proposition 2.1(3)]). Thus, compressing r more if necessary, we further obtain that $e\mathcal {Q} e r \vee r (\mathcal {Q} '\cap {\mathcal M})r e \subseteq f {\mathcal M} f$ is a finite index inclusion of II $_1$ factors, as desired.

In this section, we show that various direct products of wreath-like groups give rise to $W^*$ -superrigid groups. To show this, we first establish a product rigidity result in the same spirit of [CdSS15] (see also [CD-AD20, Dr20] for more recent similar results).

Theorem 5.2. For every $1\leq k\leq n$ , let $\Gamma _k\in \mathcal W\mathcal R (A_k, B_k \curvearrowright I_k)$ be property (T) groups where $A_k$ is Abelian, $B_k$ is an icc subgroup of a hyperbolic group and $B_k \curvearrowright I_k$ has amenable stabilizers. Denote $\Gamma = \Gamma _1\times \dots \times \Gamma _n$ , and assume that $t>0$ is a scalar and $\Lambda $ is an arbitrary group satisfying ${\mathcal M}={\mathcal L}(\Gamma )^t={\mathcal L}(\Lambda )$ .

Then one can find a product decomposition $\Lambda = \Lambda _1\times \dots \times \Lambda _n$ , some scalars $t_1,\dots ,t_n>0$ with $t_1\cdots t_n=t$ and a unitary $u\in \mathscr {U}({\mathcal M})$ so that ${\mathcal L}(\Gamma _{i})^{t_i}=u{\mathcal L}(\Lambda _i) u^*$ for any $1\leq i\leq n $ .

Proof. Without any loss of generality, we can assume that $t=1$ , as the other cases do not hide any difficulties. Throughout the proof, we denote by $\Gamma _S=\times _{i\in S}\Gamma _i$ the subproduct supported on a subset $S\subset \{1,\dots , n\}$ . We also denote by $\widehat S$ the complement of S inside $\{1,\dots , n\}$ .

Let $\Delta : {\mathcal M}\rightarrow {\mathcal M}\bar \otimes {\mathcal M}$ be the $\ast $ -embedding given by $\Delta (v_h)=v_h\otimes v_h$ for all $h\in \Lambda $ . Let $1\leq i,j\leq n $ , and observe that $\Delta ({\mathcal L}(\Gamma _{\hat i})), \Delta (\mathcal L(\Gamma _i))\subset {\mathcal M} \bar \otimes \mathcal L(\Gamma _{\hat j})\bar \otimes \mathcal L(\Gamma _j)$ are commuting subalgebras. Using Proposition 4.1 (see also [CIOS21, Theorem 6.15]), we have either $\Delta ({\mathcal L}(\Gamma _{\hat i}))\prec _{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes {\mathcal L}(\Gamma _{\hat j})\bar \otimes L(A^{(I_j)}_j)$ or $\Delta ({\mathcal L}(\Gamma _{i}))\prec _{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes {\mathcal L}(\Gamma _{\hat j})\bar \otimes {\mathcal L}(A^{(I_j)}_j)$ . Since ${\mathcal L}(A^{(I_j)}_j)$ is Abelian and $\Delta ({\mathcal L}(\Gamma _{\hat i}))$ , $\Delta ({\mathcal L}(\Gamma _{i}))$ have property (T), the prior intertwining relations further imply that either $\Delta ({\mathcal L}(\Gamma _{\hat i}))\prec _{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes {\mathcal L}(\Gamma _{\hat j})$ or $\Delta (\mathcal L(\Gamma _{i}))\prec _{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes \mathcal L(\Gamma _{\hat j})$ . Moreover, using [DHI16, Lemma 2.4(3)], we have either $\Delta (\mathcal L(\Gamma _{\hat i}))\prec ^s_{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes \mathcal L(\Gamma _{\hat j})$ or $\Delta (\mathcal L(\Gamma _{i}))\prec ^s_{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes \mathcal L(\Gamma _{\hat j})$ . If the former would hold for all $1\leq j\leq n$ , then by using [DHI16, Lemma 2.8(2)], we would get $\Delta ({\mathcal L}(\Gamma _{i}))\prec _{{\mathcal M}\bar \otimes {\mathcal M}} \bigcap ^n_{j=1} ({\mathcal M}\bar \otimes {\mathcal L}(\Gamma _{\hat j})) ={\mathcal M}\otimes 1$ . This contradicts [IPV10, Lemma 7.2]. Hence, for every $1\leq i\leq n$ , there is $1\leq j\leq n$ so that $\Delta (\mathcal L(\Gamma _{\hat i}))\prec _{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes \mathcal L(\Gamma _{\hat j})$ . Furthermore, using [Dr20, Theorem 4.3], we actually have that $\Delta (\mathcal L(\Gamma _{\hat i}))\prec _{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes \mathcal L(\Gamma _{\hat i})$ for all $1\leq i\leq n$ . Now, since $\mathcal L(\Gamma _i)$ has property (T), Theorem 6.1 shows that there is a subgroup $\Sigma <\Lambda $ , such that

(5.10) $$ \begin{align} {\mathcal L}(\Gamma_{\hat i})\prec_{\mathcal M} {\mathcal L}(\Sigma)\text{ and }{\mathcal L}(\Gamma_{i})\prec_{\mathcal M} {\mathcal L}(C_\Lambda(\Sigma)). \end{align} $$

Since ${\mathcal L}(\Gamma _i)$ is solid, (5.10) allows to apply Proposition 5.1 and find a projection ${0\neq p= qr}$ with $q\in {\mathcal L}(\Sigma ), r \in {\mathcal L}(\Sigma )'\cap {\mathcal M}$ , such that $p({\mathcal L}(\Sigma )\vee ({\mathcal L}(\Sigma )'\cap {\mathcal M}))p \subseteq p{\mathcal M} p$ is a finite index inclusion of II $_1$ factors. In particular, $p{\mathcal L}(\Sigma \mathrm {vC}_\Lambda (\Sigma ))p \subseteq p{\mathcal M} p$ is also a finite index inclusion of von Neumann algebras, and thus, $[\Lambda : \Sigma \mathrm {vC}_\Lambda (\Sigma )]<\infty $ . Since $\Lambda $ is icc property (T), then so is $\Sigma \mathrm {vC}_\Lambda (\Sigma )$ . Now, observe that one can find an increasing sequence of groups $\cdots \leqslant \Omega _n \leqslant \Omega _{n+1}\leqslant \cdots \leqslant \mathrm {vC}_\Lambda (\Sigma )$ normalized by $\Sigma $ with $\cup _n \Omega _n=\mathrm {vC}_\Lambda (\Sigma )$ whose centralizers form a descending sequence $\Sigma \geqslant C_\Sigma (\Omega _1)\geqslant \cdots \geqslant C_\Sigma (\Omega _n )\geqslant C_\Sigma (\Omega _{n+1}) \cdots $ of finite index subgroups. Therefore, $\Sigma \Omega _n \nearrow \Sigma \mathrm {vC}_\Lambda (\Sigma )$ , and using property (T), there is n so that ${\Sigma \Omega _n = \Sigma \mathrm {vC}_\Lambda (\Sigma )}$ . Since $C_\Sigma (\Omega _n) \Omega _n \leqslant \Sigma \Omega _n$ has finite index, we conclude that $C_\Lambda (\Omega _n) \Omega _n\leq \Lambda $ has finite index as well. Denote $\Sigma _0=C_\Lambda (\Omega _n)$ , and note that $C_\Lambda (\Sigma )\leq C_\Lambda (\Sigma _0)$ . From (5.10), it follows that

(5.11) $$ \begin{align} {\mathcal L}(\Gamma_{\hat i})\prec_{\mathcal M} {\mathcal L}(\Sigma_0)\text{ and }{\mathcal L}(\Gamma_{i})\prec_{\mathcal M} {\mathcal L}(C_\Lambda(\Sigma_0)). \end{align} $$

By passing to relative commutants in (5.11), we get that ${\mathcal L}(\Sigma _0)\prec _{\mathcal M} \mathcal L(\Gamma _{\hat i})$ and $\mathcal L(C_{\Lambda }(\Sigma _0))\prec _{\mathcal M} \mathcal L(\Gamma _i)$ . Since $[\Lambda :\Sigma _0 C_\Lambda (\Sigma _0)]<\infty $ , we derive that $\mathcal L(\Sigma C_\Lambda (\Sigma ))'\cap {\mathcal M}=\mathbb C 1$ . Therefore, using [DHI16, Lemma 2.4(3)], we see that

(5.12) $$ \begin{align} {\mathcal L}(\Sigma_0)\prec^s_{\mathcal M} {\mathcal L}(\Gamma_{\hat i}) \text{ and } {\mathcal L}(C_{\Lambda}(\Sigma_0))\prec^s_{\mathcal M} {\mathcal L}(\Gamma_i). \end{align} $$

By using (5.11), (5.12) together with [DHI16, Theorem 6.1] and a standard inductive argument, we obtain the desired conclusion.

Corollary 5.3. For every $1\leq k\leq n$ , let $\Gamma _k\in \mathcal W\mathcal R (A_k, B_k \curvearrowright I_k)$ be property (T) groups where $A_k$ is Abelian, $B_k$ is an icc subgroup of a hyperbolic group, $B_k \curvearrowright I_k$ has amenable stabilizers and the set $\{i\in I \; | \; g\cdot i\neq i\}$ is infinite for any $g\in B_k\setminus \{1\}$ . Denote $\Gamma = \Gamma _1\times \dots \times \Gamma _n$ , and assume that $t>0$ is any scalar, $\Lambda $ is an arbitrary group and $\theta : {\mathcal L}(\Gamma )^t\rightarrow {\mathcal L}(\Lambda )$ is any $\ast $ -isomorphism.

Then $t=1$ , and one can find a character $\eta :\Gamma \to \mathbb T$ , a group isomorphism $\delta :\Gamma \to \Lambda $ and a unitary $u\in {\mathcal L}(\Lambda )$ , such that $\theta (u_g)= \eta (g) u v_{\delta (g)} u^*$ , for all $g\in \Gamma $ .

Proof. This follows directly from Theorem 5.2 and [CIOS21, Theorem 9.1].

6 Reconstruction of clique subgroups under $W^*$ -equivalence

Towards establishing superrigidity results for graph products, the first major step is to identify in the mystery group $\Lambda $ collections of subgroups that play the same role as the full subgroups associated to clique subgraphs in the source group $\Gamma $ . This will be achieved using the commultiplication map [Io10, PV11] in combination with an ultrapower method from [Io11] (see also [DHI16, CdSS15]) and a technique for reconstructing malnormal group structure developed in [CD-AD20, CI17]. Our result is a new manifestation of a more conceptual principle called peripheral reconstruction $W^*$ -method, which consists of exploiting the natural tension that occurs between ‘a peripheral structure’ and a ‘direct product structure’ in the group. For completeness, we include all the details.

We start by recalling an ultrapower technique which is essentially contained in the proof of [Io11, Theorem 3.1] (see also [CdSS15, Theorem 3.3]) and the statement that will be used is a particular case of [DHI16, Theorem 4.1].

Theorem 6.1 [Io11].

Let $\Lambda $ be a countable icc group and denote by ${\mathcal M}={\mathcal L}(\Lambda )$ . Let $\Delta :{\mathcal M}\to {\mathcal M}\bar \otimes {\mathcal M}$ be the $*$ -homomorphism given by $\Delta (v_\lambda )=v_\lambda \otimes v_\lambda $ , for all $\lambda \in \Lambda .$ Let $\mathcal P,\mathcal {Q}\subset {\mathcal M}$ be von Neumann subalgebras, such that $\Delta (\mathcal P)\prec _{{\mathcal M}\bar \otimes {\mathcal M}}{\mathcal M}\bar \otimes \mathcal {Q}$ .

Then there exists a decreasing sequence of subgroups $\Sigma _k<\Lambda $ , such that $\mathcal P\prec _{\mathcal M} {\mathcal L}(\Sigma _k)$ , for every $k\ge 1$ , and $\mathcal {Q}'\cap {\mathcal M}\prec _{\mathcal M} {\mathcal L}(\cup _{k\ge 1} C_\Lambda (\Sigma _k)).$

We continue with the following result, which contains a consequence of Theorem 6.1 and solidity type result to certain von Neumann algebras of graph product groups. Throughout this article, if $\mathscr {C}\in \mathrm {cliq} (\mathscr {G})$ is a clique of a graph, then for every $v\in \mathscr {C}$ , we denote by $\hat v= \mathscr {C}\setminus \{v\}$ .

Theorem 6.2. Let $\Gamma = \mathscr {G} \{\Gamma _v\}$ be a graph product of groups, such that $\mathscr {G}\in \mathrm {CC}_1$ , and denote ${\mathcal M}={\mathcal L}(\Gamma )$ . Assume that for any $v\in \mathscr V$ , $\Gamma _v\in \mathcal W\mathcal R (A_v, B_v \curvearrowright I_v)$ where $A_v$ is Abelian, $B_v$ is an icc subgroup of a hyperbolic group.

Let $\Lambda $ be an arbitrary group, such that ${\mathcal M}={\mathcal L}(\Lambda )$ , and let $\mathscr {C} \in \mathrm {cliq} (\mathscr G)$ and $v\in \mathscr {C}$ . Then the following hold:

1. 1. There is a subgroup $\Lambda _{\hat v} < \Lambda $ , such that $C_\Lambda (\Lambda _{\hat v})$ is nonamenable and ${\mathcal L}(\Gamma _{\hat v})\prec _{{\mathcal M}} L(\Lambda _{\hat v})$ .

2. 2. If $\mathcal P, \mathcal {Q}, \mathcal R\subset p{\mathcal L}(\Gamma _{\mathscr {C}}) p$ are commuting von Neumann subalgebras, such that $\mathcal {Q}$ has no amenable direct summand and $\mathcal R$ is isomorphic to a corner of ${\mathcal L}(\Gamma _{\hat v})$ , then $\mathcal P$ is completely atomic.

Proof. 1. Following [IPV10, Io10], consider the $\ast $ -embedding $\Delta : {\mathcal M} \rightarrow {\mathcal M} \bar \otimes {\mathcal M}$ given by $\Delta (v_h)=v_h\otimes v_h$ for all $h\in \Lambda $ . Fix $\mathscr {C} \in \mathrm {cliq} (\mathscr G)$ . From assumptions, we have $\Delta ({\mathcal L}(\Gamma _{\mathscr {C}}))\subseteq {\mathcal M} \bar \otimes {\mathcal M} = \mathcal L(\Gamma \times \Gamma )$ is a property (T) von Neumann algebra. Thus, applying Theorem 2.8, one can find cliques $\mathscr {C}_1,\mathscr {C}_2\in \mathrm {cliq}(\mathscr {G})$ , such that $\Delta (\mathcal L(\Gamma _{\mathscr {C}}))\prec _{{\mathcal M} \bar \otimes {\mathcal M}} \mathcal L(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2})$ . This implies that one find projections $q\in \Delta (\mathcal L(\Gamma _{\mathscr {C}})) p\in \mathcal L(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2})$ , partial isometry ${w\in {\mathcal M}\bar \otimes {\mathcal M} }$ and a $\ast $ -isomorphism onto its image $\theta : q\Delta (\mathcal L(\Gamma _{\mathscr {C}})) q\rightarrow \mathcal R:= \theta (q\Delta (\mathcal L(\Gamma _{\mathscr {C}})) q)\subseteq p \mathcal L(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2}) p$ , such that $\theta (x)w=wx$ for all $x\in q \Delta (\mathcal L(\Gamma _{\mathscr {C}})) q$ . Notice that $ww^*\in \mathcal R'\cap q({\mathcal M} \bar \otimes {\mathcal M})q $ and $w^*w\in (\Delta ({\mathcal L}(\Gamma _{\mathscr {C}}))'\cap {\mathcal M}\bar \otimes {\mathcal M} )q$ . Moreover, we can assume without loss of generality that the support of $E_{\mathcal L(\Gamma _{\mathscr {C}_1} \times \Gamma _{\mathscr {C}_2})}(ww^*)$ equals p.

Since $\Gamma _t, t\in \mathscr V$ , are icc, we can assume without loss of generality that $q\in \Delta ({\mathcal L}(\Gamma _{v_0}))$ for some $v_0\in \mathscr {C}$ . For every $\mathscr {D} \subseteq \mathscr {C}$ , we denote $\mathcal R_{\mathscr {D}}= \theta (q\Delta (\mathcal L(\Gamma _{\mathscr {D}}))q)$ . Thus, $\mathcal R= \vee _{t\in \mathscr {C}} \mathcal R_v$ where $\mathcal R_t= \theta (q\Delta (\mathcal L(\Gamma _t))q)$ are mutually commuting nonamenable II $_1$ factors.

Now, fix $\emptyset \neq \mathscr {D} \subseteq \mathscr {C}$ , and notice that $\mathcal R_{\mathscr {D}} \vee \mathcal R_{\mathrm {lk}(\mathscr {D})} = \mathcal R\subset {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2})=:\tilde {\mathcal N}$ are commuting nonamenable factors. Since $\mathcal R_{\mathscr {D}}$ and $\mathcal R_{\mathrm {lk}(\mathscr {D})}$ are commuting property (T) algebras, then [CIOS21, Theorem 6.15] implies that for every $t \in \mathscr {C}_2$ , either $\mathcal R_{\mathscr {D}}\prec _{\tilde {\mathcal N}} {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}}\times A^{(B_t)}_t ) $ or $\mathcal R_{\mathrm {lk}(\mathscr {D})}\prec _{\tilde {\mathcal N}} {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}}\times A^{(B_t)}_t ) $ . However, since $R_{\mathscr {D}}, \mathcal R_{\mathrm { lk}(\mathscr {D})}$ have property (T) and $A_t^{(B_t)}$ is amenable, we further conclude that either $\mathcal R_{\mathscr {D}}\prec _{\tilde {\mathcal N}} {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}} )$ or $\mathcal R_{\mathrm {lk}(\mathscr {D})}\prec _{\tilde {\mathcal N}} {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}} )$ , and using factoriality, we actually have either a) $\mathcal R_{\mathscr {D}}\prec ^s {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}} )$ or b) $\mathcal R_{\mathrm {lk}(\mathscr {D})}\prec ^s {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}} )$ . Assume by contradiction that b) holds for all $t\in \mathscr {C}_2$ . By [DHI16, Lemma 2.6], this would imply that $\mathcal R_{\mathrm {lk}(\mathscr {D})}\prec \cap _{t\in \mathscr {C}_2} {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}} )= {\mathcal L}(\Gamma _{\mathscr {C}_1})\otimes 1$ . Using [IPV10, Proposition 7.2], this entails that $\mathcal R_{\mathrm {lk}(\mathscr {D})}$ is atomic, a contradiction. Hence, there is $t\in \mathscr {C}_2$ , such that $\mathcal R_{\mathscr {D}}\prec ^s {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{v\}} )$ . Combining this with the first part, we get again that $\Delta ({\mathcal L}(\Gamma _{\mathscr {D}}))\prec {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}}) )$ . The conclusion of the first part follows from Theorem 6.1 by letting $\mathscr {D}=\mathscr {C}\setminus \{v\}$ .

2. The conclusion follows from Proposition 4.5.

Next, by using similar methods to [CD-AD20, Theorem 9.1], we identify up to corners the clique subgroups in the mystery subgroup.

Theorem 6.3. Let $\Gamma = \mathscr {G} \{\Gamma _v\}$ be a graph product of icc groups, such that $\mathscr {G}\in \mathrm {CC}_1$ , and denote ${\mathcal M}={\mathcal L}(\Gamma )$ . Let $\mathscr {C} \in \mathrm {cliq} (\mathscr G)$ and $v\in \mathscr {C}$ with $\mathrm {lk}( \hat {v})=\{v\}$ . Assume that whenever $\mathcal P, \mathcal {Q}, \mathcal R\subset p{\mathcal L}(\Gamma _{\mathscr {C}}) p$ are commuting von Neumann subalgebras, such that $\mathcal {Q}$ has no amenable direct summand and $\mathcal R$ is isomorphic to a corner of ${\mathcal L}(\Gamma _{\hat v})$ , then $\mathcal P$ is completely atomic.

Let $\Lambda $ be an arbitrary group, such that ${\mathcal M}={\mathcal L}(\Lambda )$ , and assume that there exists a subgroup $\Lambda _{\hat v}< \Lambda $ with nonamenable centralizer $C_\Lambda (\Lambda _{\hat v})$ , such that ${\mathcal L}(\Gamma _{\hat v})\prec _{{\mathcal M}} {\mathcal L}(\Lambda _{\hat v})$ .

Then there exist a subgroup $\Lambda _{\hat v} C_\Lambda (\Lambda _{\hat v}) \leqslant \Sigma _{\mathscr {C}}<\Lambda $ with $[\Sigma : \Lambda _{\hat v} \mathrm {vC}_\Lambda (\Lambda _{\hat v})]<\infty $ and $\mathrm { QN}^{(1)}_\Lambda (\Sigma _{\mathscr {C}})=\Sigma _{\mathscr {C}}$ , a nonzero projection $c\in \mathcal Z({\mathcal L}(\Sigma _{\mathscr {C}}))$ and $w_0\in \mathscr {U}({\mathcal M})$ with $w_0c w^*_0=n\in \mathcal L(\Gamma _{\mathscr {C}})$ , such that $w_0 \mathcal L(\Sigma _{\mathscr {C}})c w^*_0= n\mathcal L (\Gamma _{\mathscr {C}})n$ .

Proof. Since ${\mathcal L}(\Gamma _{\hat v})\prec _{\mathcal M} {\mathcal L}(\Lambda _{\hat v})$ , one can find projections $a\in {\mathcal L}(\Gamma _{\hat v})$ , $f\in {\mathcal L}(\Lambda _{\hat v})$ , a nonzero partial isometry $v\in f{\mathcal M} a$ and a $\ast $ -isomorphism onto its image $\phi :a \mathcal L(\Gamma _{\hat v})a \rightarrow \mathcal B:=\phi (a\mathcal L(\Gamma _{\hat v})a) \subseteq f\mathcal L(\Lambda _{\hat v})f $ , such that

(6.1) $$ \begin{align} \phi(x)v=vx \text{ for all } x\in a{\mathcal L}(\Gamma_{\hat v})a. \end{align} $$

Notice that $vv^*\in \mathcal B'\cap f{\mathcal M} f$ and $v^*v\in a{\mathcal L}(\Gamma _{\hat v})a'\cap a{\mathcal M} a$ . Since the virtual centralizer satisfies $\mathrm {vC}_\Gamma (\Gamma _{\hat v})= C_\Gamma (\Gamma _{\hat v}) = \Gamma _v$ , then ${\mathcal L}(\Gamma _{\hat v})'\cap {\mathcal M} = {\mathcal L}(\Gamma _v)$ . Thus, we can write ${v^*v=aa_0}$ for a projection $a_0\in {\mathcal L}(\Gamma _v)$ . Equation (6.1) implies that $\mathcal B vv^* = v {\mathcal L}(\Gamma _{\hat v})v^* = u_1 {\mathcal L}(\Gamma _{\hat v}) v^*v u^*_1$ , where $u_1\in {\mathcal M}$ is a unitary extending v. Taking relative commutants, we get $vv^*(\mathcal B' \cap f{\mathcal M} f)vv^*= u_1 v^*v( a{\mathcal L}(\Gamma _{\hat v})a'\cap a{\mathcal M} a )v^*v u^*_1= u_1 v^*v\mathcal L(\Gamma _{v} ) v^*v u^*_1$ . Hence, $vv^*(\mathcal B \vee \mathcal B'\cap f{\mathcal M} f)vv^*=\mathcal B vv^*\vee vv^*(\mathcal B' \cap f{\mathcal M} f)vv^*= u_1 v^*v\mathcal L(\Gamma _{\mathscr {C}}) v^*vu^*_1$ . Therefore, since $\mathcal L(\Gamma _{\mathscr {C}})$ is a factor, one can find a new unitary $u_2 \in \mathscr {U}({\mathcal M})$ , such that

(6.2) $$ \begin{align} (\mathcal B \vee \mathcal B'\cap f{\mathcal M} f) z_2 \subseteq u_2 {\mathcal L}(\Gamma_{\mathscr{C}}) u^*_2, \end{align} $$

here, $z_2$ is the central support of $vv^*$ in $\mathcal B \vee \mathcal B'\cap f{\mathcal M} f$ . In particular, we have $z_2\in \mathcal Z(\mathcal B'\cap f{\mathcal M} f )$ and $vv^*\leq z_2 \leq f$ .

Now, observe that (6.2) implies that

(6.3) $$ \begin{align} {\mathcal L}(C_\Lambda(\Lambda_{\hat v}))z_2\subseteq (f{\mathcal L}(\Lambda_{\hat v} )f'\cap f {\mathcal M} f )z_2\subseteq (\mathcal B'\cap f {\mathcal M} f)z_2\subseteq u_2 {\mathcal L}(\Gamma_{\mathscr{C}})u_2^*. \end{align} $$

Next, we will show the following containment

(6.4) $$ \begin{align}z_2({\mathcal L}(C_\Lambda(\Lambda_{\hat v}))f'\cap f{\mathcal M} f) z_2\subseteq u_2\mathcal L(\Gamma_{\mathscr{C}})u_2^*.\end{align} $$

Since we have ${\mathcal M}= u_2{\mathcal L}(\Gamma _{\mathscr V\setminus \{v\}} \ast _{\Gamma _{\hat v}} \Gamma _{\mathscr {C}} ) u^*_2=u_2{\mathcal L}(\Gamma _{\mathscr V\setminus \{v\}})u_2^* \ast _{u_2{\mathcal L}(\Gamma _{\hat v})u_2^*} u_2\mathcal L (\Gamma _{\mathscr {C}} ) u^*_2$ , then by [IPP05, Theorem 1.2.1] and (6.3) to get (6.4), it suffices to show that

(6.5) $$ \begin{align}\mathcal L(C_\Lambda(\Lambda_{\hat v}))z_2\nprec_{u_2{\mathcal L} (\Gamma_{\mathscr{C}} ) u^*_2} u_2{\mathcal L}(\Gamma_{\hat v})u_2^*.\end{align} $$

Since $\Gamma _v$ is icc, it follows that $vv^*(\mathcal B' \kern1.3pt{\cap}\kern1.3pt f{\mathcal M} f)vv^*{=} u_1 v^*v{\mathcal L}(\Gamma _{v} ) v^*v u^*_1{=} u_1 a_0(\mathcal L(\Gamma _{v} ) )a_0 a u^*_1$ is a factor. Since $z_2$ is the central support of $vv^*$ in $\mathcal B' \cap f{\mathcal M} f$ , then $(\mathcal B' \cap f{\mathcal M} f) z_2$ is a factor as well.

Assume that (6.5) does not hold. Thus, one can find projections $t\in {\mathcal L}(C_\Lambda (\Lambda _{\hat v})), r\in u_2 \mathcal L(\Gamma _{\hat v})u_2^*$ , a partial isometry $w\in ru_2{\mathcal L}(\Gamma _{\mathscr {C}})u_2^* tz_2$ and a $\ast $ -isomorphism onto its image $\psi : t{\mathcal L}(C_\Lambda (\Lambda _{\hat v}))tz_2\rightarrow ru_2\mathcal L(\Gamma _{\hat v})u_2^* r$ , such that

(6.6) $$ \begin{align} \psi(x)w=wx \text{ for all } x\in t{\mathcal L}(C_\Lambda(\Lambda_{\hat v}))tz_2. \end{align} $$

Notice that $t{\mathcal L}(C_\Lambda (\Lambda _{\hat v}))tz_2\subseteq t(\mathcal B'\cap f {\mathcal M} f)tz_2\subseteq u_2 {\mathcal L}(\Gamma _{\mathscr {C}})u_2^*$ , and observe one can pick t small enough so that $tz_2$ is subequivalent to $v^*v$ inside $(\mathcal B'\cap f {\mathcal M} f)z_2$ . Using this, one can find a unitary $u_3\in {\mathcal M}$ , such that $t{\mathcal L}(C_\Lambda (\Lambda _{\hat v}))tz_2\subseteq u_3 v^*v \mathcal L(\Gamma _v)v^*v u_3^*$ . Hence, using this in combination with relation (6.6), we see that for all unitaries $x\in t\mathcal L(C_\Lambda (\Lambda _{\hat v}))tz_2$ , we have that $\psi (x)$ is a unitary in $r u_2 \mathcal L(\Gamma _{\mathscr {C}}) u_2^* r$ , and therefore,

(6.7) $$ \begin{align} \begin{aligned} \|E_{u_2 {\mathcal L}(\Gamma_{\hat v}) u_2^*}(ww^*)\|_2&=\|E_{u_2 {\mathcal L}(\Gamma_{\hat v}) u_2^*}(\psi(x)ww^*)\|_2= \| E_{u_2{\mathcal L}(\Gamma_{\hat v})u_2^*}(wxw^*)\|_2\\ &=\| E_{u_2{\mathcal L}(\Gamma_{\hat v})u_2^*}(wE_{u_3 {\mathcal L}(\Gamma_v)u_3^*}(x)w^*)\|_2\\ &=\| E_{{\mathcal L}(\Gamma_{\hat v})}(u_2^*w u_3E_{{\mathcal L}(\Gamma_v)}(u_3^*x u_3)u_3^*w^* u_2)\|_2. \end{aligned} \end{align} $$

Since $w\neq 0$ , then (6.7) and basic approximations of $u_2^*w u_3$ and $u_3$ imply the existence of a finite subset $F\subset \Gamma $ and a constant $C>0$ , such that for all unitaries $x\in t{\mathcal L}(C_\Lambda (\Lambda _{\hat v}))tz_2$ , we have

(6.8) $$ \begin{align} \sum_{g,h,k,l\in F }| E_{{\mathcal L}(\Gamma_{\hat v})}(u_gE_{{\mathcal L}(\Gamma_v)}(u_hx u_k) u_l)\|_2\geq C. \end{align} $$

However, we see that for all $x\in t{\mathcal L}(C_\Lambda (\Lambda _{\hat v}))tz_2$ , we have

$$ \begin{align*}\begin{aligned}\sum_{g,h,k,l\in F }| E_{{\mathcal L}(\Gamma_{\hat v})}(u_gE_{{\mathcal L}(\Gamma_v)}(u_hx u_k) u_l)\|_2&= \sum_{g,h,k,l\in F }| P_{ g^{-1}\Gamma_{\hat v}l^{-1}\cap \Gamma_v}(u_hx u_k)\|_2 \\&= \sum_{g,h,k,l\in F, gl^{-1}\in \Gamma_v }| P_{ (g^{-1}\Gamma_{\hat v}g^{-1}\cap \Gamma_v)gl^{-1}}(u_hx u_k)\|_2\\&= \sum_{g,h,k,l\in F, gl^{-1}\in \Gamma_v }| E_{ {\mathcal L}(g^{-1}\Gamma_{\hat v}g^{-1}\cap \Gamma_v)}(u_hx u_{klg^{-1}})\|_2\\ &=\sum_{g,h,k,l\in F, gl^{-1}\in \Gamma_v }| \tau(x u_{klg^{-1}h})|.\end{aligned}\end{align*} $$

Here, we denoted by $P_S:\ell ^2(\Gamma )\to \ell ^2(\Gamma )$ the orthogonal projection onto the $\|\cdot \|_2$ -closure of the $\mathrm {span}\{u_g|g\in S\}$ for any subset $S\subset \Gamma $ . As F is finite and $t{\mathcal L}(C_\Lambda (\Lambda _{\hat v}))tz_2$ is diffuse, there is a sequence of unitaries $x_n \in t\mathcal L(C_\Lambda (\Lambda _{\hat v}))tz_2$ , such that $\sum _{g,h,k,l\in F, gl^{-1}\in \Gamma _v }| \tau (x_n u_{klg^{-1}h})|\rightarrow 0$ as $n\rightarrow \infty $ . This, however, contradicts (6.8). Hence, (6.5) must hold.

Thus, (6.4) implies that $z_2 ({\mathcal L}(C_\Lambda (\Lambda _{\hat v}))f\vee {\mathcal L}(C_\Lambda (\Lambda _{\hat v}))f'\cap f{\mathcal M} f) z_2\subseteq u_2\mathcal L(\Gamma _{\mathscr {C}})u_2^*$ . Again, since $\mathcal L(\Gamma _{\mathscr {C}})$ is a factor, there is $u\in \mathscr {U}({\mathcal M})$ so that

(6.9) $$ \begin{align} ({\mathcal L}(C_\Lambda(\Lambda_{\hat v}))f\vee {\mathcal L}(C_\Lambda(\Lambda_{\hat v}))f'\cap f{\mathcal M} f) z\subseteq u\mathcal L(\Gamma_{\mathscr{C}})u^*,\end{align} $$

where z is the central support of $z_2$ in $(\mathcal L(C_\Lambda (\Lambda _{\hat v}))f\vee \mathcal L(C_\Lambda (\Lambda _{\hat v}))f'\cap f{\mathcal M} f)$ . In particular, we have $vv^*\leq z_2 \leq z \leq f$ . Now since $f\mathcal L(\Lambda _{\hat v} )f\subseteq \mathcal L(C_\Lambda (\Lambda _{\hat v}))f'\cap f{\mathcal M} f$ , then by (6.9), we get $(f{\mathcal L}(\Lambda _{\hat v} )f\vee {\mathcal L}(C_\Lambda (\Lambda _{\hat v}))f)z\subseteq u{\mathcal L}(\Gamma _{\mathscr {C}})u^*$ , and hence

(6.10) $$ \begin{align} u^*({\mathcal L}(C_\Lambda(\Lambda_{\hat v}))f\vee f {\mathcal L}(\Lambda_{\hat v} )f) z u\subseteq {\mathcal L}(\Gamma_{\mathscr{C}}).\end{align} $$

In particular, (6.10) implies that $u^*f{\mathcal L}(\Lambda _{\hat v})fzu\subseteq {\mathcal L}(\Gamma _{\mathscr {C}})$ . Since $vv^*\leq z\in f{\mathcal L}(C_\Lambda (\Lambda _{\hat v})) f'\cap f{\mathcal M} f$ and $\mathcal B$ is a factor, then the map $\phi ': a\mathcal L(\Gamma _{\hat v})a \rightarrow u^* \mathcal B z u \subseteq u^*f \mathcal L(\Lambda _{\hat v})fz u$ given by $\phi '(x)=u^* \phi (x)z u$ still defines a $\ast $ -isomorphism that satisfies $\phi '(x) w=wx$ , for any $x\in a\mathcal L(\Gamma _{\hat v})a$ , where $w= u^*zv$ is a nonzero partial isometry. Hence, $\mathcal L(\Gamma _{\hat v}) \prec _{{\mathcal M}} u^*f\mathcal L(\Lambda _{\hat v})fzu$ . Thus, by Corollary 2.17, it follows that $\mathcal L(\Gamma _{\hat v})\prec _{\mathcal L(\Gamma _{\mathscr {C}})} u^*f\mathcal L(\Lambda _{\hat v})fzu$ .

To this end, using [CKP14, Proposition 2.4] and its proof, we can find nonzero ${p \in \mathscr P({\mathcal L}(\Gamma _{\hat v}))}$ , $r=u^*ezu \in u^*f{\mathcal L}(\Lambda _{\hat v})fzu$ with $e\in \mathscr P(f{\mathcal L}(\Lambda _{\hat v})f)$ , a von Neumann subalgebra $\mathcal C \subseteq u^*e\mathcal L(\Lambda _{\hat v})ezu $ and a $\ast $ -isomorphism $\theta : p\mathcal L(\Gamma _{\hat v})p \rightarrow \mathcal C$ , such that:

1. a) the inclusion $\mathcal C \vee (\mathcal C' \cap u^*e{\mathcal L}(\Lambda _{\hat v})ezu ) \subseteq u^*e{\mathcal L}(\Lambda _{\hat v})ezu $ has finite index;

2. b) there is a nonzero partial isometry $y \in {\mathcal L}(\Gamma _{\mathscr {C}})$ , such that $\theta (x) y =y x$ for all ${x\in p{\mathcal L}(\Gamma _{\hat v})p}$ , where $y^*y\in p{\mathcal L}(\Gamma _{\hat v})p'\cap p{\mathcal M} p$ and $yy^*\in \mathcal C '\cap r{\mathcal M} r$ .

Note that $r\in {\mathcal L}(\Gamma _{\mathscr {C}})$ and $\mathcal C$ , $\mathcal C' \cap u^* r{\mathcal L}(\Lambda _{\hat v})r z u$ and $u^*\mathcal L(C_\Lambda (\Lambda _{\hat v}))r z u$ are commuting von Neumann subalgebras of $r {\mathcal L}(\Gamma _{\mathscr {C}})r$ . Since $\mathcal C$ is isomorphic to a corner of $\mathcal L(\Gamma _{\hat v})$ and $u^*\mathcal L(C_\Lambda (\Lambda _{\hat v}))ezu$ has no amenable direct summand, then the assumption implies that $\mathcal C' \cap u^* e\mathcal L(\Lambda _{\hat v})ezu$ is purely atomic. Thus, one can find a nonzero projection $q \in \mathcal Z(\mathcal C' \cap u^* e{\mathcal L}(\Lambda _{\hat v})e zu )$ , such that after compressing the containment in a) by q and replacing $\mathcal C$ by $\mathcal C q$ , y by $qy$ and $\theta (x)$ by $\theta (x) q$ in b), we can assume in addition that $\mathcal C \subseteq u^*e{\mathcal L}(\Lambda _{\hat v})ezu $ is a finite index inclusion of nonamenable II $_1$ factors. By [PP86, Proposition 1.3], it follows that $\mathcal C \subseteq u^*e{\mathcal L}(\Lambda _{\hat v})ezu $ admits a finite Pimsner-Popa basis, which implies that there exist $x_1,\dots ,x_m\in u^*e{\mathcal L}(\Lambda _{\hat v})ezu$ , such that $u^*e{\mathcal L}(\Lambda _{\hat v})ezu=\sum _{i=1}^m x_i \mathcal C$ . Note also that $u^*e{\mathcal L}(\Lambda _{\hat v})ezu \subset r{\mathcal M} r$ since $r=u^*ezu$ . Hence,

(6.11) $$ \begin{align} \mathcal {QN}_{ r{\mathcal M} r}(\mathcal C ) "=\mathcal {QN}_{ r{\mathcal M} r}( u^*e{\mathcal L}(\Lambda_{\hat v})ezu )". \end{align} $$

Also, the intertwining relation in b) shows that $\mathcal C yy^*= y p{\mathcal L}(\Gamma _{\hat v})p y^*= l p{\mathcal L}(\Gamma _{\hat v})p y^*y l$ where $yy^*\in \mathcal C'\cap r {\mathcal M} r$ and $l\in {\mathcal L}(\Gamma _{\mathscr {C}})$ is a unitary extending y, meaning $y=ly^*y$ . Therefore, using the quasi-normalizer formulas for group von Neumann algebras and for compressions, Lemmas 2.15 and 2.13 (and Remark 2.14), respectively, we deduce that

(6.12) $$ \begin{align}\begin{aligned} ly^*y {\mathcal L}(\Gamma_{\mathscr{C}}) y^*yl^* & = ly^*y {\mathcal L}(\mathrm{QN}_\Gamma (\Gamma_{\hat v})) y^*yl^* \overset{L.2.15}{=} l y^*y \mathcal{QN}_{{\mathcal M}}({\mathcal L}(\Gamma_{\hat v}))"y^*yl^*\\ & \overset{L.2.13}{=}\mathcal {QN}_{l y^*y {\mathcal M} y^*yl}(l p{\mathcal L}(\Gamma_{\hat v})p y^*y l)"= \mathcal {QN}_{yy^*{\mathcal M} yy^*}(\mathcal C yy^*)" \\ &\overset{R.2.14}{=}yy^*\mathcal {QN}_{ r{\mathcal M} r}(\mathcal C ) "yy^*\overset{(6.11)}{=}yy^*\mathcal {QN}_{ r{\mathcal M} r}( u^* e{\mathcal L}(\Lambda_{\hat v})ezu )" yy^*\\ &\overset{L.2.13}{=} yy^* u^*z e\mathcal {QN}_{ {\mathcal L}(\Lambda)}({\mathcal L}(\Lambda_{\hat v}) )"ez u yy^*\overset{L.2.15}{=}yy^* u^* z e{\mathcal L} (\mathrm{QN}_{\Lambda}(\Lambda_{\hat v}) )e zu yy^*. \end{aligned}\end{align} $$

Denote $\Upsilon = \mathrm {QN}_\Lambda (\Lambda _{\hat v})$ and $\Sigma = \langle \mathrm {QN}^{(1)}_\Lambda (\Upsilon ))\rangle <\Lambda $ . As $\mathrm {QN}^{(1)}_\Gamma (\Gamma _{\mathscr {C}})= \Gamma _{\mathscr {C}}$ , then formula (6.12) together with the corresponding formulas for one-sided quasi-normalizers, Lemmas 2.13 and 2.15, show that

(6.13) $$ \begin{align}yy^* u^* ze{\mathcal L} (\Upsilon)e zu yy^* =yy^* u^* z e{\mathcal L} (\Sigma)ez u yy^*= ly^*y {\mathcal L}(\Gamma_{\mathscr{C}}) y^*yl^*.\end{align} $$

In particular, by [CI17, Lemma 2.2], we have $[\Sigma :\Upsilon ]<\infty $ , and one can also check that $\mathrm {QN}^{(1)}_\Lambda (\Sigma )=\Sigma $ .

Notice the above relations also show that $yy^*= u^* d u$ for some projection $d\in ze{\mathcal L} (\Sigma )ez$ . Thus, relation (6.13) entails that $ u^* d{\mathcal L} (\Sigma ) d u = ly^*y {\mathcal L}(\Gamma _{\mathscr {C}}) y^*yl^*$ , and letting $w_0:=ul \in \mathscr {U}({\mathcal M})$ and $t=y^*y$ , we conclude that $w^*_0 d {\mathcal L}(\Sigma )d w_0= t {\mathcal L}(\Gamma _{\mathscr {C}}) t$ . Moreover, if we replace $w^*_0 \Sigma w_0$ by $\Sigma $ and we use $w^*_0 d w_0= t$ , we have that $t {\mathcal L}(\Sigma )t = t {\mathcal L}(\Gamma _{\mathscr {C}})t$ . As $\mathcal L(\Gamma _{\mathscr {C}})$ is a factor, one can find a unitary $w_1 \in {\mathcal M}$ so that if c denotes the central support of $t \in {\mathcal L}(\Sigma )$ , we have that ${\mathcal L}(\Sigma ) c \subseteq w_1\mathcal L(\Gamma _{\mathscr {C}})w_1^*$ . This implies that there exists a projection $h \in {\mathcal L}(\Gamma _{\mathscr {C}})$ , such that $t= w_1 hw_1^*$ . Moreover, since ${\mathcal L}(\Gamma _{\mathscr {C}})$ is a factor, there is a unitary $w_2 \in \mathcal L(\Gamma _{\mathscr {C}})$ so that $t = w_2 hw^*_2$ . Altogether, these relations show that $wt=tw$ where $w :=w_1 w_2^*$ . Also, we have that ${\mathcal L}(\Sigma )c\subseteq w {\mathcal L}(\Gamma _{\mathscr {C}})w^*$ . Multiplying on both sides by t, we get $t {\mathcal L}(\Gamma _{\mathscr {C}})t=t\mathcal L(\Sigma )t\subseteq t w\mathcal L(\Gamma _{\mathscr {C}})w^* t$ , and hence $tw^*t \mathcal L(\Gamma _{\mathscr {C}}) t\subseteq t \mathcal L(\Gamma _{\mathscr {C}})t w^* t$ . In particular, using Corollary 2.17, we get $w^*t =tw^*t\in \mathcal {QN}^{(1)}_{t{\mathcal M} t}(t{\mathcal L}(\Gamma _{\mathscr {C}})t)=t {\mathcal L}(\Gamma _{\mathscr {C}})t$ , and hence $wt \in t {\mathcal L}(\Gamma _{\mathscr {C}})t$ . Altogether, these relations imply that $t {\mathcal L}(\Sigma )t= t{\mathcal L}(\Gamma _{\mathscr {C}})t=w t \mathcal L(\Gamma _{\mathscr {C}})t w^*$ . Since $\mathcal L(\Sigma ) c\subseteq w \mathcal L(\Gamma _{\mathscr {C}})w^*$ , we apply the ‘moreover’ part in [CI17, Lemma 2.6] and derive that $\mathcal L(\Sigma ) c= c \mathcal L(\Gamma _{\mathscr {C}})c$ .

In conclusion, we showed there is a subgroup $\Lambda _{\hat v} \mathrm {C}_\Lambda (\Lambda _{\hat v})<\mathrm {QN}_\Lambda (\Lambda _{\hat v})<\Sigma <\Lambda $ , such that $[\Sigma :\mathrm {QN}_\Lambda (\Lambda _{\hat v})]<\infty $ , and $\mathrm {QN}^{(1)}_\Lambda (\Sigma )=\Sigma $ . Moreover, there are $r\in \mathscr P(\mathscr Z({\mathcal L}(\Sigma )))$ , $v_0\in \mathscr {U}({\mathcal M})$ and $n\in \mathscr P(\mathcal L(\Gamma _{\mathscr {C}}))$ so that

(6.14) $$ \begin{align} v_0{\mathcal L}(\Sigma) rv^*_0=c{\mathcal L}(\Gamma_{\mathscr{C}})c.\end{align} $$

Finally, we show that $[\Sigma : \Lambda _{\hat v} \mathrm {vC}_\Lambda (\Lambda _{\hat v})]<\infty $ . Let $u^*qzu=s\in u^* e {\mathcal L}(\Lambda _{\hat v})e z u$ be a projection, and let $\mathcal C_1\subseteq \mathcal C \subseteq u^* e {\mathcal L}(\Lambda _{\hat v})e z u$ be a subfactor, such that $u^* e {\mathcal L}(\Lambda _{\hat v})e z u=\langle \mathcal C, s \rangle $ is the basic construction for $\mathcal C_1\subseteq \mathcal C$ . Notice that ${\mathcal D}_1 := \theta ^{-1}(\mathcal C_1)\subseteq p\mathcal L(\Gamma _{\hat v})p$ is a finite index subfactor. Let $\Theta : {\mathcal D}_1 \rightarrow \mathcal C_1s= u^*q \mathcal L(\Lambda _{\hat v})qzu$ be the isomorphism given by $\Theta (x)=\theta (x)s$ for all $x\in {\mathcal D}_1$ . Using b), we have that $\Theta (x) sy=sy x$ for all $x\in {\mathcal D}_1$ . Since s is a Jones’s projection for $\mathcal C_1\subseteq \mathcal C$ , one can check that $sy\neq 0$ . Letting w be the polar decomposition of $sy$ , we further have that $\Theta (x) w=w x$ for all $x\in {\mathcal D}_1$ . By Corollary 2.17, we have $s\in {\mathcal L}(\Gamma _{\mathscr {C}})$ , $ww^*\in u^*q\mathcal L(\Lambda _{\hat v})qzu'\cap s{\mathcal M} s \subset s\mathcal L(\Gamma _{\mathscr {C}})s$ and $w^*w \in p{\mathcal D}_1 p' \cap p{\mathcal M} p\subset p{\mathcal D}_1 p' \cap pL(\Gamma _{\mathscr {C}}) p$ . Letting $w_o$ be a unitary extending w and using the previous intertwining relation, we see that $u^* q{\mathcal L}(\Lambda _{\hat v})qzu ww^*= w_o {\mathcal D}_1 w^*_o$ . By taking relative commutants, we obtain that $ ww^*(u^* q\mathcal L(\Lambda _{\hat v})qzu'\cap s {\mathcal M} s)ww^*=w_o {\mathcal D}_1'\cap p\mathcal L(\Gamma _{\mathscr {C}})p w_o^* $ , and hence,

(6.15) $$ \begin{align}\begin{aligned} &w_o ({\mathcal D}_1\vee \mathcal L(\Gamma_v)p )w_o^*\subset w_o ({\mathcal D}_1\vee {\mathcal D}_1'\cap p\mathcal L(\Gamma_{\mathscr{C}})p )w_o^*=\\ &=ww^*(u^* q{\mathcal L}(\Lambda_{\hat v})qzu \vee u^* qL(\Lambda_{\hat v})qzu'\cap s {\mathcal M} s)ww^* \\ &=ww^*u^* qz ({\mathcal L}(\Lambda_{\hat v}) \vee {\mathcal L}(\Lambda_{\hat v})'\cap {\mathcal M} ) qzu ww^*\subseteq ww^* {\mathcal L}(\Gamma_{\mathscr{C}})ww^*.\end{aligned}\end{align} $$

Since $\Gamma _{\mathscr {C}}$ is malnormal in $\Gamma $ , the containment $w_o ({\mathcal D}_1\vee {\mathcal L}(\Gamma _v)p )w_o^*\subseteq ww^* {\mathcal L}(\Gamma _{\mathscr {C}})ww^*$ implies that $w=w_o p\in {\mathcal L}(\Gamma _{\mathscr {C}})$ , and hence $ww^* {\mathcal L}(\Gamma _{\mathscr {C}})ww^*= w{\mathcal L}(\Gamma _{\mathscr {C}})w^*= w_o p{\mathcal L}(\Gamma _{\mathscr {C}})pw_o^*$ . Therefore, since ${\mathcal D}_1\vee {\mathcal L}(\Gamma _v)p\subseteq p\mathcal L(\Gamma _{\mathscr {C}})p$ has finite index, then so does $w_o ({\mathcal D}_1\vee \mathcal L(\Gamma _v)p )w_o^*\subset ww^*\mathcal L(\Gamma _{\mathscr {C}}) ww^*$ . In particular, all inclusions in (6.15) are of finite index. Hence, we have ${\mathcal L}(\Gamma _{\mathscr {C}})\prec _{\mathcal M} {\mathcal L}(\Lambda _{\hat v})\vee \mathcal L(\Lambda _{\hat v})'\cap {\mathcal M}$ and since $\mathcal L(\Lambda _{\hat v})\vee \mathcal L(\Lambda _{\hat v})'\cap {\mathcal M}\subseteq \mathcal L(\Lambda _{\hat v} vC_\Lambda (\Lambda _{\hat v}))$ , we get $\mathcal L(\Gamma _{\mathscr {C}})\prec _{\mathcal M} \mathcal L(\Lambda _{\hat v} vC_\Lambda (\Lambda _{\hat v}))$ . Using (6.14), this further implies that $\mathcal L(\Sigma )\prec _{\mathcal M} \mathcal L(\Lambda _{\hat v} \mathrm { vC}_\Lambda (\Lambda _{\hat v}))$ . Moreover, since $\Lambda _{\hat v}\mathrm {vC}_\Lambda (\Lambda _{\hat v})<\Sigma $ and $\mathrm {QN}^{(1)}(\Sigma )=\Sigma $ , we actually have that ${\mathcal L}(\Sigma )\prec _{{\mathcal L}(\Sigma )} {\mathcal L}(\Lambda _{\hat v} \mathrm {vC}_\Lambda (\Lambda _{\hat v}))$ . Using [DHI16, Lemma 2.5], this implies that $[\Sigma : \Lambda _{\hat v} \mathrm {vC}_\Lambda (\Lambda _{\hat v})]<\infty $ , as claimed.

We continue with the following technical result which goes back to [CI17, Theorem 3.2]. The proof goes along the same lines with the proof of [CD-AD20, Theorem 9.2], and we include all the details for completeness.

Theorem 6.4. Assume that $\Gamma = \mathscr {G} \{\Gamma _v\}$ is a graph product of groups, such that $\mathscr {G}\in \mathrm {CC}_1$ and all vertex groups $\Gamma _v$ are icc, nonamenable. Let $\mathscr {C}_1,\dots ,\mathscr {C}_n$ be an enumeration of its consecutive cliques, and denote $\Gamma _i=\Gamma _{\mathscr {C}_i}$ , for any $i\in \overline {1,n}$ .

Let $\Lambda $ be an arbitrary group, such that ${\mathcal M}={\mathcal L}(\Gamma )=\mathcal L(\Lambda )$ . Assume that for any ${i\in \{1,\dots ,n\}}$ , there exist $\Lambda _i<\Lambda $ with $\mathrm {QN_\Lambda ^{(1)}}(\Lambda _i)=\Lambda _i$ and a subset $i\in J_i\subset \overline {1,n}$ satisfying:

1. 1. For any $k\in J_i$ , there exists a projection $0\neq z_i^k\in \mathcal Z({\mathcal L}(\Lambda _i))$ , such that $\sum _{k\in J _i}z_i^k=1$ ;

2. 2. For any $k\in J_i$ , there exists $u_i^k\in \mathcal U({\mathcal M})$ , such that:

   1. a. $u_i^i {\mathcal L}(\Lambda _i) z_i^i (u_i^i)^*=p_i{\mathcal L}(\Gamma _i)p_i$ , where $p_i=u_i^i z_i^i (u_i^i)^*$ ;

   2. b. $u_i^k {\mathcal L}(\Lambda _i) z_i^k (u_i^k)^*\subset {\mathcal L}(\Gamma _k)$ .

Then there exists a partition $T_1\sqcup \dots \sqcup T_l= \{1,...,n\}$ and a subgroup $\tilde \Lambda _i<\Lambda $ with $\mathrm {QN_\Lambda ^{(1)}}(\tilde \Lambda _i)=\tilde \Lambda _i$ for any $1\leq i\leq l$ , such that:

1. 1. For any $k\in T_i$ , there exists a projection $0\neq \tilde z_i^k\in \mathcal Z({\mathcal L}(\tilde \Lambda _i))$ satisfying $\sum _{k\in T _i}\tilde z_i^k=1$ ;

2. 2. For any $k\in T_i$ , there exists $\tilde u_i^k\in \mathcal U({\mathcal M})$ , such that $\tilde u_i^k {\mathcal L}(\tilde \Lambda _i) \tilde z_i^k (\tilde u_i^k)^* = \tilde p_k^i{\mathcal L}(\Gamma _k) \tilde p_k^i$ , where $\tilde p_k^i=\tilde u_i^k \tilde z_i^k (\tilde u_i^k)^*$ .

Proof. We fix an arbitrary $k\in J_1\setminus \{1\}$ . Note that the assumption 2. implies that ${\mathcal L}(\Lambda _1)\prec _{{\mathcal M}} \mathcal L (\Gamma _k)$ and $\mathcal L(\Gamma _k)\prec ^s_{{\mathcal M}} \mathcal L(\Lambda _k)$ since $\mathcal L(\Gamma _k)'\cap {\mathcal M}=\mathbb C1.$ Using [Va08, Lemma 3.7], we deduce that ${\mathcal L}(\Lambda _1)\prec _{{\mathcal M}} {\mathcal L}(\Lambda _k)$ . Using [CI17, Lemma 2.2], there exists $h_k\in \Lambda $ , such that $[\Lambda _1: h_k\Lambda _kh_k^{-1}\cap \Lambda _1]<\infty $ . Up to replacing $\Lambda _k$ by $h_k \Lambda _k h^{-1}_k$ , we may assume that $h_k=1$ .

We continue by showing that for all $i\in J_1, j\in J_k$ with $i\neq j$ , we have $z_1^i z_k^j=0$ . By assuming the contrary, we consider some $i\in J_1,j\in J_k$ with $i\neq j$ satisfying $z_1^i z_k^j\neq 0$ . Assumption 2. gives that ${\mathcal L}(\Lambda _k\cap \Lambda _1) z_1^i\prec _{{\mathcal M}}^s {\mathcal L}(\Gamma _i)$ and ${\mathcal L}(\Lambda _k\cap \Lambda _1) z_k^j\prec _{{\mathcal M}}^s {\mathcal L}(\Gamma _j)$ . From [DHI16, Lemma 2.4], there exist maximal projections $a_1^i, a_k^j\in \mathcal Z({\mathcal L}(\Lambda _k\cap \Lambda _1)'\cap {\mathcal M})$ , such that ${\mathcal L}(\Lambda _k\cap \Lambda _1) a_1^i\prec _{{\mathcal M}}^s {\mathcal L}(\Gamma _i)$ and ${\mathcal L}(\Lambda _k\cap \Lambda _1) a_k^j\prec _{{\mathcal M}}^s {\mathcal L}(\Gamma _j)$ . Since $z_1^i z_k^j\neq 0$ , we deduce that $a_1^i a_k^j\in \mathcal Z({\mathcal L}(\Lambda _k\cap \Lambda _1)'\cap {\mathcal M})$ is a nonzero projection. Thus, we can apply [Va10a, Lemma 2.7] to get that ${\mathcal L}(\Lambda _k\cap \Lambda _1) a_1^i a_k^j\prec _{{\mathcal M}}^s {\mathcal L}(\Gamma _i\cap g \Gamma _j g^{-1})$ for some $g\in \Gamma $ . Since $[\Lambda _1: \Lambda _k\cap \Lambda _1]<\infty $ , we therefore deduce that ${\mathcal L}(\Gamma _1)\prec _{{\mathcal M}} {\mathcal L}(\Lambda _k\cap \Lambda _1)$ and ${\mathcal L}(\Lambda _k\cap \Lambda _1)\prec _{{\mathcal M}} \mathcal L(\Gamma _{\hat v})$ for some $v\in \mathscr {C}_{i}$ . By Lemma 2.10, we get that $\Gamma _v$ is amenable, contradiction.

Thus, we have $J_k=J_1$ and $z_1^i=z_k^i$ for all $i\in J_1$ . Indeed, if $i\in J_1$ , then there exists $l\in J_k$ , such that $z_1^i z_k^l\neq 0$ . The previous paragraph shows that $l=i$ . This shows, in particular, that $J_1\subset J_k$ . By symmetry, we also get that $J_k\subset J_1$ . The previous paragraph also implies that $z_1^i z_k^{i'}= 0$ for all $i'\in J_k$ with $i'\neq i$ . This implies that $z_1^i\leq z_k^i$ , and the conclusion follows again by symmetry reasons.

Next, we show that $\Lambda _k=\Lambda _1$ . Assumption 2. gives that $\text {for all } i\in J_1=J_k$ , we have

(6.16) $$ \begin{align} u_1^i {\mathcal L}(\Lambda_k\cap \Lambda_1)z_1^i (u_1^i)^*\subset {\mathcal L}(\Gamma_i) \text{ and } u_k^i {\mathcal L}(\Lambda_k\cap \Lambda_1)z_k^i (u_k^i)^*\subset {\mathcal L}(\Gamma_i). \end{align} $$

Fix an element $w_i\in \mathscr {C}_i$ , and note that $\Gamma = \Gamma _{\mathscr V\setminus \{w_i\}} \ast _{\Gamma _{\hat {w_i}}} \Gamma _i$ , where $\Gamma =\mathscr {G}\{\Gamma _v,v \in \mathscr V\}$ . As before, Lemma 2.10 implies that

(6.17) $$ \begin{align} u_k^i {\mathcal L}(\Lambda_k\cap \Lambda_1)z_k^i (u_k^i)^* \nprec_{{\mathcal L}(\Gamma_i)} {\mathcal L}(\Gamma_{\hat {w_i}}). \end{align} $$

By letting $a^i=u_1^i z_1^i (u_k^i)^*$ , we note that

(6.18) $$ \begin{align} a^i u_k^i {\mathcal L}(\Lambda_k\cap \Lambda_1)z_k^i (u_k^i)^*= u_1^i {\mathcal L}(\Lambda_k\cap \Lambda_1) z_1^i (u_1^i)^* a^i. \end{align} $$

By using (6.16), (6.17), (6.18) and by applying [IPP05, Theorem 1.2.1], we get that $a^i\in {\mathcal L}(\Gamma _i)$ . Assumption 2. implies that

$$ \begin{align*}u_1^1 {\mathcal L}(\Lambda_1) z_1^1 (u_1^1)^*=p_1{\mathcal L}(\Gamma_1)p_1 \text{ and } u_k^1 {\mathcal L}(\Lambda_k) z_k^1 (u_k^1)^*\subset {\mathcal L}(\Gamma_1). \end{align*} $$

By noticing that $z_1^1=z_k^1$ and by conjugating the previous inclusion by $a^1\in {\mathcal L}(\Gamma _1)$ , we derive that

$$ \begin{align*}u_1^1 {\mathcal L}(\Lambda_1) z_1^1 (u_1^1)^*=p_1{\mathcal L}(\Gamma_1)p_1 \text{ and } u_1^1 {\mathcal L}(\Lambda_k) z_1^1 (u_1^1)^*\subset {\mathcal L}(\Gamma_1). \end{align*} $$

Hence, $u_1^1 z_1^1 (u_1^1)^*=p_1$ , and therefore, ${\mathcal L}(\Lambda _k\cap \Lambda _1)z_1^1\subset {\mathcal L}(\Lambda _k)z_1^1\subset \mathcal L(\Lambda _1)z_1^1$ . Since ${[\Lambda _1: \Lambda _k\cap \Lambda _1]<\infty }$ , we deduce that $\mathcal L(\Lambda _k\cap \Lambda _1)z_1^1\subset \mathcal L(\Lambda _k)z_1^1$ admits a finite Pimsner-Popa basis. By applying [CdSS15, Proposition 2.6], we derive that $[\Lambda _k: \Lambda _k\cap \Lambda _1]<\infty $ . In combination with $[\Lambda _1: \Lambda _k\cap \Lambda _1]<\infty $ , we get that

$$ \begin{align*}\Lambda_k=\mathrm{QN}^{(1)}_\Lambda (\Lambda_k)= \mathrm{QN}^{(1)}_\Lambda (\Lambda_k\cap\Lambda_1)=\mathrm{QN}^{(1)}_\Lambda (\Lambda_1)= \Lambda_1. \end{align*} $$

Therefore, we have $u_1^1{\mathcal L}(\Lambda _1) z_1^1 (u_1^1)^*=p_1\mathcal L(\Gamma _1) p_1$ and $u_k^k\mathcal L(\Lambda _1) z_k^k (u_k^k)^*=p_k\mathcal L(\Gamma _k) p_k$ . Since $z_1^k=z_k^k$ and all these work for all $k\in J_1$ , we get the conclusion of the proof for the first element of the partition. Namely, we let $\tilde \Lambda _1=\Lambda _1$ , $T_1=J_1$ , $\tilde u_1^k= u_k^k $ and $\tilde z_1^k= z_k^k$ . Finally, if $T_1=\{1,\dots ,n\}$ , then the proof is completed. Otherwise, pick an element $t\in \{1,\dots ,n\}\setminus T_1$ , and repeat the above arguments, starting with $J_t$ .

Theorem 6.5. Let $\Gamma = \mathscr {G} \{\Gamma _v\}$ be a graph product of groups, such that $\mathscr {G}\in \mathrm {CC}_1$ and for any $v\in \mathscr V$ , $\Gamma _v\in \mathcal W\mathcal R (A_v, B_v \curvearrowright I_v)$ where $A_v$ is Abelian, $B_v$ is an icc subgroup of a hyperbolic group. Assume that $|\mathscr {C}|\neq |\mathscr {D}|$ for any two distinct cliques $\mathscr {C},\mathscr {D}\in \mathrm {cliq}(\mathscr {G})$ .

Let $\Lambda $ be an arbitrary group, such that ${\mathcal L}(\Gamma )={\mathcal L}(\Lambda )$ . Thus, for every clique $\mathscr {C}\in \mathrm {cliq}( \mathscr {G})$ , there exist a unitary $u_{\mathscr {C}} \in {\mathcal M}$ and a subgroup $\Lambda _{\mathscr {C}} \leqslant \Lambda $ , such that $u_{\mathscr {C}} \mathcal L(\Gamma _{\mathscr {C}}) u_{\mathscr {C}}^*= \mathcal L(\Lambda _{\mathscr {C}})$ .

Proof. The proof is using several techniques from [CI17, CD-AD20]. First, let $\mathscr {C}_1,\dots ,\mathscr {C}_n$ be an enumeration of its consecutive cliques, and denote $\Gamma _i=\Gamma _{\mathscr {C}_i}$ , for any $1\leq i\leq n$ .

Claim 6.6. For any $1\leq i\leq n$ , there exist $\Lambda _i<\Lambda $ with $\mathrm {QN_\Lambda ^{(1)}}(\Lambda _i)=\Lambda _i$ and a subset $i\in J_i\subset \{1,...,n\}$ satisfying:

1. 1. For any $k\in J_i$ , there exists a projection $0\neq z_i^k\in \mathcal Z({\mathcal L}(\Lambda _i))$ , such that $\sum _{k\in J _i}z_i^k=1$ ;

2. 2. For any $k\in J_i$ , there exists $u_i^k\in \mathcal U({\mathcal M})$ , such that:

   1. a. $u_i^i {\mathcal L}(\Lambda _i) z_i^i (u_i^i)^*=p_i{\mathcal L}(\Gamma _i)p_i$ , where $p_i=u_i^i z_i^i (u_i^i)^*$ ;

   2. b. $u_i^k {\mathcal L}(\Lambda _i) z_i^k (u_i^k)^*\subset {\mathcal L}(\Gamma _k)$ .

Proof of the Claim 6.6.

Fix $1\leq i\leq n$ . By applying Theorems 6.2 and 6.3, there exist a subgroup $\Lambda _i<\Lambda $ with $\mathrm {QN_\Lambda ^{(1)}}(\Lambda _i)=\Lambda _i$ , a projection $0\neq z_i\in \mathcal Z({\mathcal L}(\Lambda _i))$ and $u_i\in \mathscr {U}({\mathcal M})$ with $p_i=u_iz_i (u_i)^*\in {\mathcal L} (\Gamma _i)$ , such that

(6.19) $$ \begin{align} u_i{\mathcal L}(\Lambda_i)z_i (u_i)^*= p_i {\mathcal L}(\Gamma_i) p_i. \end{align} $$

Since $\Gamma _i$ has property (T), then (6.19) together with [CI17, Lemma 2.13] imply that $\Lambda _i$ has property (T) as well. Using $\mathrm {QN_\Lambda ^{(1)}}(\Lambda _i)=\Lambda _i$ and ${\mathcal L}(\Lambda _i)'\cap {\mathcal M}\subset {\mathcal L}(\mathrm {vC}_{\Lambda }(\Lambda _i))$ , it follows that

(6.20) $$ \begin{align} {\mathcal L}(\Lambda_i)'\cap {\mathcal M}=\mathscr Z ({\mathcal L}(\Lambda_i)). \end{align} $$

Since ${\mathcal L}(\Gamma _j)$ is a II $_1$ factor for any $1\leq j\leq n$ , by using (6.20), there is a maximal projection $z_j^i\in \mathscr Z({\mathcal L}(\Lambda _i))$ , such that ${\mathcal L}(\Lambda _i)z_j^i$ can be unitarily conjugated into $\mathcal L(\Gamma _j)$ ; hence, we take $u_j^i\in \mathscr {U}({\mathcal M})$ , such that $u_j^i\mathcal L(\Lambda _i)z_j^i (u_j^i)^*\subset \mathcal L(\Gamma _j).$ We continue by showing that $\bigvee _{j=1}^n z_j^i=1$ . By assuming the contrary, then $z=1-\bigvee _{j=1}^n z_j^i$ is a nonzero projection of $\mathscr Z({\mathcal L}(\Lambda _i))$ . Since ${\mathcal L}(\Lambda _i)z$ has property (T), Theorem 2.8 implies that there exists $j\in \overline {1,n}$ , such that ${\mathcal L}(\Lambda _i)z\prec _{{\mathcal M}} \mathcal L(\Gamma _j)$ . Note that $\mathcal L(\Lambda _i)z\nprec _{{\mathcal M}} \mathcal L(\Gamma _{\mathscr {C}_j\setminus \{v\}})$ , for any $v\in \mathscr {C}_j$ . Indeed, otherwise, there is $v\in \mathscr {C}_j$ , such that $\mathcal L(\Lambda _i)\prec _{{\mathcal M}} \mathcal L(\Gamma _{\mathscr {C}_j\setminus \{v\}})$ . By using [Va08, Lemma 3.5] and (6.20), we get ${\mathcal L}(\Gamma _{v})\prec _{{\mathcal M}} \mathcal L(\Lambda _i)'\cap {\mathcal M}=\mathscr Z(\mathcal L(\Lambda _i))$ , which is impossible since $\Gamma _v$ is nonamenable. Therefore, the ‘moreover’ part of Theorem 2.8 gives a nonzero projection $z_0\in \mathscr Z (\mathcal L(\Lambda _i))$ and $u_0\in \mathscr {U}({\mathcal M})$ , such that $u_0 \mathcal L(\Lambda _i)z_0 u_0^*\subset \mathcal L(\Gamma _j)$ . Recall that $u_j^i{\mathcal L}(\Lambda _i)z_j^i (u_j^i)^*\subset {\mathcal L}(\Gamma _j).$ Since ${\mathcal L}(\Gamma _j)$ is a II $_1$ factor, we can perturb $u_0$ by a different unitary and assume that $u_0z_0 u_0^*$ and $u_j^iz_j^i (u_j^i)^*$ are orthogonal projections in $\mathcal L(\Gamma _j)$ . By letting $u=u_0z_0+u_j^iz_j^i$ , we note that $u^*u=z_0+z_j^i$ and let $v\in \mathscr {U}({\mathcal M})$ , such that $vu^*u=u$ . Therefore, $v \mathcal L(\Lambda _i)(z_0+z_j^i) v^*\subset \mathcal L(\Gamma _j)$ , which contradicts the maximality of $z_j^i$ . This shows that $\bigvee _{j=1}^n z_j^i=1$ .

Next, we prove that $z_i^i=z_i$ . By assuming the contrary, the maximality of $z_i^i$ implies that $c:=z_i^i-z_i$ is a nonzero projection in $\mathscr Z({\mathcal L}(\Lambda _i))$ . By using (6.19) and $u_i^i{\mathcal L}(\Lambda _i)c (u_i^i)^*\subset {\mathcal L}(\Gamma _j)$ , we can proceed as in the previous paragraph and assume that $p_i$ and $p:=u_i^i c (u_i^i)^*$ are orthogonal projections of ${\mathcal L}(\Gamma _i)$ . Hence, by letting $u=u_i^i c + u_i z_i$ , we derive that $uu^*=p+p_i$ , $u^*u=c+z_i$ and

(6.21) $$ \begin{align} u{\mathcal L}(\Lambda_i)(c+z_i) u^*\subset (p+p_i){\mathcal L}(\Gamma_j)(p+p_i). \end{align} $$

Since $\mathrm {QN_\Lambda ^{(1)}}(\Lambda _i)=\Lambda _i$ , we obtain from Lemmas 2.13 and 2.15 that

(6.22) $$ \begin{align} \mathrm{\mathscr{QN}}^{(1)}_{(p+p_i){\mathcal L}(\Gamma_j)(p+p_i)}(u{\mathcal L}(\Lambda_i)(c+z_i) u^*)=u{\mathcal L}(\Lambda_i)(c+z_i) u^*. \end{align} $$

Relations (6.19), (6.21) and (6.22) and the fact that $(p+p_i){\mathcal L}(\Gamma _j)(p+p_i)$ is a II $_1$ factor allow to apply the ‘moreover’ part of [CI17, Lemma 2.6] and deduce that $u{\mathcal L}(\Lambda _i)(c+z_i) u^*= (p+p_i){\mathcal L}(\Gamma _j)(p+p_i).$ This is a contradiction, since the center of $u\mathcal L(\Lambda _i)(c+z_i) u^*$ is at least two-dimensional. This shows that $z_i^i=z_i$ .

Finally, since $\bigvee _{j=1}^n z_j^i=1$ , for any $1\leq i\leq n$ , we can eventually replace $z_j^i$ by a smaller projection in $\mathscr Z({\mathcal L}(\Lambda _i))$ and assume that $\{z_j^i\}_{j=1}^n$ are mutually orthogonal. The claim follows by letting $J_i=\{1\leq k\leq n\,:\, z_k^i\neq 0 \}$ .

Next, we can apply Theorem 6.4 and obtain a partition there exist a partition $T_1\sqcup \dots \sqcup T_l= \{1,...,n\}$ and a subgroup $\tilde \Lambda _i<\Lambda $ with $\mathrm {QN_\Lambda ^{(1)}}(\tilde \Lambda _i)=\tilde \Lambda _i$ for any $1\leq i\leq l$ , such that:

1. 1. For any $k\in T_i$ , there exists a projection $0\neq \tilde z_i^k\in \mathcal Z({\mathcal L}(\tilde \Lambda _i))$ satisfying $\sum _{k\in T _i}\tilde z_i^k=1$ ;

2. 2. For any $k\in T_i$ , there exists $\tilde u_i^k\in \mathcal U({\mathcal M})$ , such that $\tilde u_i^k {\mathcal L}(\tilde \Lambda _i) \tilde z_i^k (\tilde u_i^k)^* = \tilde p_k^i{\mathcal L}(\Gamma _k) \tilde p_k^i$ , where $\tilde p_k^i=\tilde u_i^k \tilde z_i^k (\tilde u_i^k)^*$ .

Note that it is enough to show that $|T_i|=1$ , for all $1\leq i\leq l$ . Hence, we fix an arbitrary $1\leq i\leq l$ and assume by contradiction that $|T_i|>2$ . Take $k,j \in T_i$ , two distinct elements, such that $|\mathscr {C}_k|>|\mathscr {C}_j|$ . Note that we have ${\mathcal L}(\Gamma _k)\prec _{{\mathcal M}} {\mathcal L}(\tilde \Lambda _i)$ and $\mathcal L(\tilde \Lambda _i)\prec _{{\mathcal M}} \mathcal L(\Gamma _j)$ . Proposition 4.4 implies that $|\mathscr {C}_k|\leq |\mathscr {C}_j|$ , contradiction. This shows that $|T_i|=1$ , for all $1\leq i\leq l$ , and the conclusion follows.

7 Superrigidity results within the category of graph product groups

In this final section, we derive a strong rigidity result for factors arising from graph product groups considered in Section 5 (see Theorem 7.6). Building upon the results in the prior sections, we show that the graph product groups considered in Theorem 6.5 are completely recognizable from the category of all von Neumann algebras associated to any nontrivial graph product group with infinite vertex groups. Notice that unlike previous strong rigidity results [IPP05, CH08], in our case, we do not need any other additional a priori assumptions on the vertex groups or on the underlying graph.

To prove our theorem, we first establish a result which asserts that the click subgroups of the graph products from Theorem 6.5 along with various other aspects of their ‘position’ in the ambient group are features which are completely recognizable from the von Neumann algebra framework. The result relies heavily on Theorem 6.5, and for the proof, we only explain how it follows from this. Although, for deriving Theorem 7.6 we do not need all the parts of the conclusion, we decided to state it here in its complete and somewhat technical form as this could be instrumental for further attempts to establish $W^*$ -superrigidity of these groups.

Theorem 7.1. Let $\mathscr {G}\in \mathrm {CC}_1$ , let $\mathrm {cliq}(\mathscr {G})=\{ \mathscr {C}_1, \ldots , \mathscr {C}_n\}$ be a consecutive cliques enumeration and assume that $|\mathscr {C}_i|\neq |\mathscr {C}_j|$ whenever $i\neq j$ . Let $\Gamma = \mathscr {G}\{\Gamma _v\}$ be a graph product group, such that for any $v\in \mathscr V$ , $\Gamma _v\in \mathcal W\mathcal R (A_v, B_v \curvearrowright I_v)$ where $A_v$ is Abelian, $B_v$ is an icc subgroup of a hyperbolic group and the set $\{i\in I_v \; | \; g\cdot i\neq i\}$ is infinite for any $g\in B_v\setminus \{1\}$ . Denote ${\mathcal M}={\mathcal L}(\Gamma )$ .

Let $\Lambda $ be an arbitrary group, such that ${\mathcal M}={\mathcal L}(\Lambda )$ . Then for any $1\leq i\leq n$ , one can find a unitary $w_i\in {\mathcal M}$ and a subgroup $\Lambda _i<\Lambda $ satisfying the following relations:

1. 1. $\mathbb T w_i \Gamma _{\mathscr {C}_i} w_i^* =\mathbb T \Lambda _i$ for all $1\leq i\leq n$ ;

2. 2. $\mathbb T w_i \Gamma _{\mathscr {C}_{i,i+1}} w_i^*= \mathbb T w_{i+1} \Gamma _{\mathscr {C}_{i,i+1}} w_{i+1}^*= \mathbb T \Lambda _{i,i+1}$ where $\Lambda _{i,i+1}= \Lambda _i\cap \Lambda _{i+1}$ for all $1\leq i\leq n-1$ ;

3. 3. There is an $s\in \Lambda $ , such that $\mathbb T w_n \Gamma _{\mathscr {C}_{n,1}} w_n^*= \mathbb T v_s w_{1} \Gamma _{\mathscr {C}_{n,1}} w_{1}^* v_{s^{-1}}= \mathbb T \Lambda _{n}\cap s \Lambda _1 s^{-1}$ ;

4. 4. $w_i{\mathcal L}(\Gamma _{\mathscr {C}_i\cup \mathscr {C}_{i+1}})w_i^*=w_{i+1}{\mathcal L}(\Gamma _{\mathscr {C}_i\cup \mathscr {C}_{i+1}})w_{i+1}^* = {\mathcal L}(\Lambda _{i,i+1} C_\Lambda (\Lambda _{i,i+1}))$ , and hence $w^*_{i+1}w_i\in {\mathcal L} (\Gamma _{\mathscr {C}_i\cup \mathscr {C}_{i+1}})$ , for all $1\leq i\leq n-1$ ;

5. 5. $\vee ^{n-1}_{i=1} \left (\Lambda _{i,i+1} C_\Lambda (\Lambda _{i,i+1})\right )=\Lambda $ .

Proof. Using Theorem 6.5 and Corollary 5.3, one can find for any $1\leq i\leq n$ , a unitary $x_i\in {\mathcal M}$ and a subgroup $\Lambda _i<\Lambda $ satisfying $\mathbb T x_i \Gamma _{\mathscr {C}_i} x_i^* =\mathbb T \Lambda _i$ . The prior relations show that $\mathbb T x_1 \Gamma _{\mathscr {C}_{1,2}} x_1^* =\mathbb T \Lambda ^{\prime }_1$ and $\mathbb T x_2 \Gamma _{\mathscr {C}_{1,2}} x_2^* =\mathbb T \Lambda ^{\prime }_2$ for two subproduct groups $\Lambda ^{\prime }_1<\Lambda _1$ and $\Lambda ^{\prime }_2<\Lambda _2$ . Thus, for every $i=1,2$ , there exist a group isomorphism $\delta _i: \Gamma _{\mathscr {C}_{1,2}}\rightarrow \Lambda _i'$ and a multiplicative character $\eta _i :\mathscr {C}_{1,2}\to \mathbb T$ , such that

(7.1) $$ \begin{align} x_i u_g x_i^*=\eta_i(g)v_{\delta_i(g)}, \text{ for all } g\in \Gamma_{\mathscr{C}_{1,2}}. \end{align} $$

Letting $\eta (g)=\overline {\eta _2(g)}\eta _1(g)$ , one can see that altogether these relations imply that $\eta (g) x_2x_1^* v_{\delta _1(g)}= \overline {\eta _2(g)} x_2 u_g x_1^*= v_{\delta _2(g)} x_2x_1^\ast $ or all $g\in \Gamma _{\mathscr {C}_{1,2}} $ . Hence, $\eta (g) x_2x_1^* = v_{\delta _2(g)} x_2x_1^*v_{\delta _1(g^{-1})}$ , for all $g\in \Gamma _{\mathscr {C}_{1,2}} $ . If we consider the Fourier decomposition of $x_2x_1^*$ in ${\mathcal M}={\mathcal L}(\Lambda )$ , then basic analysis together with the previous relation show that $x_2x_1^*$ is supported on the set of those $h\in \Lambda $ for which the orbit $\{ \delta _2(g) h \delta _1(g^{-1}) \,:\, g\in \Gamma _{\mathscr {C}_{1,2}}\}$ is finite. This finiteness condition yields that for any such h, one can find a finite index subgroup $\Sigma _h\leqslant \Gamma _{\mathscr {C}_{1,2}}$ , such that $\delta _2(g) h \delta _1(g^{-1})=h$ , and hence, $\delta _1(g) = h^{-1}\delta _2(g)h$ , for all $g\in \Sigma _h$ . Thus, replacing $\delta _2$ by $\mathrm {ad}(h^{-1})\circ \delta _2$ , $x_2$ by $v_{h^{-1}} x_2$ and $\Lambda _2$ by $h^{-1}\Lambda _2 h$ , we can assume that ${\delta _1}_{|_{\Sigma _h}}= {\delta _2}_{|_{\Sigma _h}}$ . Moreover, using this combined with (7.1), we deduce that $u_g x u_{g^{-1}}= \eta (g) x$ where $x=x_1^*x_2$ . However, this shows that x is supported on elements $h\in \Gamma $ with finite $\Sigma _h$ -conjugation orbits. Using the icc condition and the graph product structure, we can see that such elements belong to $\Gamma _{\mathscr {C}_1\triangle \mathscr {C}_2}$ , and hence, $x\in {\mathcal L}(\Gamma _{\mathscr {C}_1\triangle \mathscr {C}_2})$ . Therefore, it follows from (7.1) that $\eta _1(g)v_{\delta _1(g)}=x_1 u_g x_1^*= x_2 u_g x_2^*= \eta _2(g)v_{\delta _2(g)}$ for all $g\in \Gamma _{\mathscr {C}_{1,2}}$ . In particular, $\delta _1=\delta _2$ on $ \Gamma _{\mathscr {C}_{1,2}}$ ; altogether, this implies the conclusion for $i=1$ . Continuing by induction, after conjugating the $\Lambda _i$ ’s by elements in $\Lambda $ , we obtain the conclusion of the first three parts.

Next, we prove 4. First notice that 2. implies

(7.2) $$ \begin{align}w_i{\mathcal L}(\Gamma_{\mathscr{C}_{i,i+1}})w_i^*=w_{i+1} {\mathcal L}(\Gamma_{\mathscr{C}_{i,i+1}}) w_{i+1}^*= {\mathcal L}(\Lambda_{i,i+1}).\end{align} $$

By using the graph product structure and by taking quasi-normalizers in (7.2), we apply Lemma 2.15 to derive that $w_i{\mathcal L}(\Gamma _{\mathscr {C}_i\cup \mathscr {C}_{i+1}})w_i^*= w_i {\mathcal L}(\mathrm {QN}_\Gamma (\Gamma _{\mathscr {C}_{i,i+1}})) w_i^*=w_{i} \mathscr {QN}_{{\mathcal M}}({\mathcal L}(\Gamma _{\mathscr {C}_{i,i+1}}))"w_i^*\kern1.3pt{=}\kern1.3ptw_{i+1} \mathscr {QN}_{{\mathcal M}}({\mathcal L}(\Gamma _{\mathscr {C}_{i,i+1}}))" w_{i+1}^*\kern1.3pt{=}\kern1.3pt w_{i+1} ({\mathcal L}(\mathrm {QN}_\Gamma (\Gamma _{\mathscr {C}_{i,i+1}})) w_{i+1}^*\kern1.3pt{=} w_{i+1} {\mathcal L}(\Gamma _{\mathscr {C}_i\cup \mathscr {C}_{i+1}}\kern-0.1pt) w_{i+1}^* $ . The same quasi-normalizer formula also implies that $w_i\mathcal L(\Gamma _{\mathscr {C}_i\cup \mathscr {C}_{i+1}}\kern-0.1pt) w_i^*=w_{i+1} \mathcal L(\Gamma _{\mathscr {C}_i\cup \mathscr {C}_{i+1}}) w_{i+1}^*= \mathcal L(\mathrm {QN}_\Lambda (\Lambda _{i,i+1})$ . In particular, $\mathcal L(\mathrm {QN}_\Lambda (\Lambda _{i,i+1})$ is a II $_1$ factor. Taking relative commutants in 2. we also have that $w_i\mathcal L(\Gamma _{\mathscr {C}_{i}\triangle \mathscr {C}_{i+1}})w_i^*=w_{i+1} \mathcal L(\Gamma _{\mathscr {C}_{i}\triangle \mathscr {C}_{i+1}}) w_{i+1}^*= \mathcal L(\Lambda _{i,i+1})'\cap \mathcal L(\Lambda )$ . Altogether, these relations imply that ${\mathcal L}(\Lambda _{i,i+1})'\cap {\mathcal L}(\Lambda ) \vee {\mathcal L}(\Lambda _{i,i+1})={\mathcal L}(\mathrm {QN}_\Lambda (\Lambda _{i,i+1})) $ . In particular, we have $\mathrm {QN}_\Lambda (\Lambda _{i,i+1})= \Lambda _i vC_\Lambda (\Lambda _{i,i+1})$ .

Moreover, factoriality also shows that $({\mathcal L}(\Lambda _{i,i+1})'\cap {\mathcal L}(\Lambda )) \bar \otimes {\mathcal L}(\Lambda _{i,i+1})={\mathcal L}(\mathrm {QN}_\Lambda (\Lambda _{i,i+1})) $ . As ${\mathcal L}(\Lambda _{i,i+1})'\cap \mathcal L(\Lambda )\subseteq \mathcal L(vC_{\Lambda }(\Lambda _{i,i+1}))$ , then Ge’s tensor splitting theorem [Ge95, Theorem A] further implies that ${\mathcal L}(vC_{\Lambda }(\Lambda _{i,i+1}))=\left ( \mathcal L(\Lambda _{i,i+1})'\cap \mathcal L(\Lambda ) \right )\bar \otimes B$ , where $B= \mathcal L(vC_{\Lambda }(\Lambda _{i,i+1}))\cap \mathcal L(\Lambda _{i,i+1})$ . However, we can see that $B = {\mathcal L}(vC_{\Lambda }(\Lambda _{i,i+1}))\cap {\mathcal L}(\Lambda _{i,i+1})={\mathcal L}(vC_{\Lambda }(\Lambda _{i,i+1})\cap \Lambda _{i,i+1})= \mathbb C 1$ , as $\Lambda _{i,i+1}$ is icc. In conclusion, we have shown that $\mathcal L(vC_{\Lambda }(\Lambda _{i,i+1}))=\mathcal L(\Lambda _{i,i+1})'\cap \mathcal L(\Lambda )$ which implies that $vC_{\Lambda }(\Lambda _{i,i+1})=C_\Lambda (\Lambda _{i,i+1})$ . Hence, $\mathrm {QN}_\Lambda (\Lambda _{i,i+1})= \Lambda _{i,i+1} C_\Lambda (\Lambda _{i,i+1})$ . Then the remaining part of 4. follows from (7.2) and Lemma 2.15 because of the graph product structure of $\mathscr {G}$ .

Part 5. follows from part 4. Indeed, since $w_2{\mathcal L}(\Gamma _{\mathscr {C}_1\cup \mathscr {C}_{2}})w_2^*= {\mathcal L}(\Lambda _{1,2} C_\Lambda (\Lambda _{1,2}))$ and $w_{2}{\mathcal L}(\Gamma _{\mathscr {C}_2\cup \mathscr {C}_{3}})w_{2}^* \kern1pt{=}\kern1pt {\mathcal L}(\Lambda _{2,3} C_\Lambda (\Lambda _{2,3}))$ , we further get $w_2 {\mathcal L}(\Gamma _{\mathscr {C}_1\cup \mathscr {C}_2\cup \mathscr {C}_{3} }) w_2^*\kern1pt{=}\kern1pt \mathcal L(\Lambda _{1,2} C_\Lambda (\Lambda _{1,2})\kern1pt{\vee} \Lambda _{2,3} C_\Lambda (\Lambda _{2,3}))$ . Since $w_3^*w_2\in \mathcal L (\Gamma _{\mathscr {C}_2\cup \mathscr {C}_3})$ , we deduce $w_3 \mathcal L(\Gamma _{\mathscr {C}_1\cup \mathscr {C}_2\cup \mathscr {C}_{3} }) w_3^*= \mathcal L(\Lambda _{1,2} C_\Lambda (\Lambda _{1,2})\vee \Lambda _{2,3} C_\Lambda (\Lambda _{2,3}))$ . Continuing in this fashion, we get the desired conclusion by induction.

Remark 7.2. We observe that, if in item 3., one could better control where the element s is located, then the statement will actually imply superrigidity of these groups. For example, this is the case if s can be picked in the subgroups $\Lambda _1$ or $\Lambda _n$ . This, however, seems difficult to establish at this time. We also note that the control of the elements s is closely related with relating the consecutive unitaries $w_i$ ’s, which, as we have seen in the previous results, is key in the reconstruction problem.

From a different perspective, if one assumes a priori that the mysterious subgroup $\Lambda $ has a nontrivial graph product structure, the analysis can be enhanced and a reconstruction statement can be obtained. Moreover, in this case, we do not even need to use the full conclusion of Theorem 7.1. Before proceeding to the result, we need three elementary lemmas.

Lemma 7.3. Let $A,B$ be countable groups, and let $C\subset \mathscr {U}({\mathcal L}(A))$ be a countable subgroup of unitaries. Assume there exists a unitary $x\in {\mathcal L}(A\times B)$ , such that $\mathbb T xCx^*< \mathbb T (A\times B)$ . Then there exists a finite index subgroup $C_0\leqslant C$ , such that $\mathbb T xC_0x^*< \mathbb T A$ .

Proof. From the assumption there exist group homomorphisms $\alpha : C\rightarrow A$ , $\beta : C\rightarrow B$ and a character $\mu \in C\to \mathbb T$ , such that for all $c\in C$ , we have

(7.3) $$ \begin{align} x c x^*= \mu(c) u_{\alpha(c)}\otimes v_{\beta(c)}. \end{align} $$

We denoted by $\{u_a\}_{a\in A}$ and $\{v_b\}_{b\in B}$ the canonical unitaries that generate ${\mathcal L}(A)$ and ${\mathcal L}(B)$ , respectively. Letting $x= \sum _{b\in B}x_b \otimes v_b$ be the Fourier expansion, where $x_b\in {\mathcal L}(A),b\in B$ , and using (7.3), we see that

$$ \begin{align*} \sum_b x_b c \otimes v_b= xc= \mu(c) u_{\alpha(c)}\otimes v_{\beta(c)} x = \sum_b \mu(c) u_{\alpha(c)} x_b \otimes v_{\beta(c) b} = \sum_b \mu(c) u_{\alpha(c)} x_{\beta(c) ^{-1}b} \otimes v_{b}, \end{align*} $$

for any $c\in C$ . Identifying the coefficients above, we get $x_b c= \mu (c) u_{\alpha (c)} x_{\beta (c) ^{-1}b}$ for all $b\in B,c\in C$ . In particular, we have $\|x_b\|_2= \| x_{\beta (c) ^{-1}b}\|_2$ , for all $b\in B,c\in C$ . Thus, if we denote the image group by $B_0 := \beta (C)<B$ , we get that $\|x_b\|_2$ is constant on each left coset $B/B_0$ . This implies that $B_0$ is finite, and letting $C_0:= \mathrm {ker} (\beta )$ , we get the desired conclusion.

Lemma 7.4. Let $\Gamma _1,\Gamma _2$ be countable groups, and assume $\Sigma <\Gamma _1\times \Gamma _2$ is a subgroup.

If there is a finite index subgroup $\Gamma _1^0<\Gamma _1$ with $\Gamma _1^0<\Sigma $ , then there is a subgroup $\Gamma _2^0<\Gamma _2$ , such that $\Gamma _1^0\times \Gamma _2^0<\Sigma $ has finite index.

Proof. Let $\pi :\Sigma \to \Gamma _1/\Gamma _1^0$ be the composition between the restriction to $\Sigma $ of the projection map $\Gamma _1\times \Gamma _2 \ni (g_1,g_2)\to g_1\in \Gamma _1$ with the quotient map $\Gamma _1\to \Gamma _1/\Gamma _1^0$ . Since the kernel of $\pi $ equals $\Sigma \cap (\Gamma _1^0\times \Gamma _2)$ and the image of $\pi $ is finite, it follows that $[\Sigma : \Sigma \cap (\Gamma _1^0\times \Gamma _2)]<\infty $ . Since $\Gamma _1^0<\Sigma $ , we get $[\Sigma : \Gamma _1^0\times (\Sigma \cap \Gamma _2)]<\infty $ .

For the following lemma, we introduce the following ad hoc definition. We say that a countable group $\Gamma $ is virtually prime if $\Gamma $ does not admit a finite index subgroup $\Gamma _0$ , which is generated by two infinite commuting subgroups. The next lemma shows that products of virtually prime groups satisfy a unique prime factorization result.

Lemma 7.5. Let $\Gamma =\Gamma _1\times \dots \times \Gamma _n$ be a product of countable groups that are virtually prime.

If $\Sigma _1\times \Sigma _2<\Gamma $ is a finite index product subgroup, then there exist finite index subgroups $\Sigma ^0_i<\Sigma _i, 1\leq i\leq 2$ , and $\Gamma ^0_j<\Gamma _j$ , $1\leq j\leq n$ , and a partition $T_1\sqcup T_2=\{1,\dots ,n\}$ , such that $\Sigma _i^0=\times _{j\in T_i}\Gamma _j^0$ , for any $1\leq i\leq 2$ .

Proof. For any $j\in \overline {1,n}$ , let $\pi _j:\Gamma \to \Gamma _j$ be the canonical projection. Since $\Gamma _j$ is virtually prime and $\pi _j(\Sigma _1)$ and $\pi _j(\Sigma _2)$ are commuting subgroups of $\Gamma _j$ , it follows that either $\pi _j(\Sigma _1)$ or $\pi _j(\Sigma _2)$ is finite. Hence, there exist finite index subgroups $\Sigma _1^0<\Sigma _1$ and $ \Sigma _2^0<\Sigma _2$ and a map $\sigma :\{1,...,n\}\to \{1,2\}$ , such that $\pi _j(\Sigma _{\sigma (j)}^0)<\Gamma _j$ has finite index and $\pi _j(\Sigma ^0_{\sigma (j)+1})=1$ for any $1\leq j\leq n$ . Here, we use the notation that $\Sigma _{3}=\Sigma _1$ . By letting $T_1=\sigma ^{-1}(1), T_2=\sigma ^{-1}(2)$ , we derive that $\Sigma _i^0=\times _{j\in T_i} \pi _j(\Sigma ^0_{i})$ , for any $1\leq i\leq 2$ , which ends the proof.

With these preparations at hand, we are now ready to state and prove our main result.

Theorem 7.6. Let $\mathscr {G}\in \mathrm {CC}_1$ , let $\mathrm {cliq}(\mathscr {G})=\{ \mathscr {C}_1, \ldots , \mathscr {C}_n\}$ be a consecutive cliques enumeration and assume that $|\mathscr {C}_i|\neq |\mathscr {C}_j|$ whenever $i\neq j$ . Let $\Gamma = \mathscr {G}\{\Gamma _v\}$ , be a graph product group, such that for any $v\in \mathscr V$ , $\Gamma _v\in \mathcal W\mathcal R (A_v, B_v \curvearrowright I_v)$ where $A_v$ is Abelian, $B_v$ is an icc subgroup of a hyperbolic group and the set $\{i\in I_v \; | \; g\cdot i\neq i\}$ is infinite for any $g\in B_v\setminus \{1\}$ .

Let $\theta : {\mathcal L}(\Gamma )\rightarrow {\mathcal L}(\Lambda )$ be any $\ast $ -isomorphism where $\Lambda $ is any nontrivial graph product group whose vertex groups are infinite.

Then, one can find a character $\eta : \Gamma \to \mathbb T$ , a group isomorphism $\delta :\Gamma \to \Lambda $ , an automorphism of ${\mathcal L}(\Lambda )$ of the form $\phi _{a,b}$ (see the notation after equation (2.2) in Section 2) and a unitary $u\in {\mathcal L}(\Lambda )$ , such that $\theta = \mathrm {ad}(u)\circ \phi _{a,b}\circ \Psi _{\eta , \delta }$ .

Proof. Assume that $\Lambda = \Lambda _{\mathscr {H}}$ , and let $\mathrm {cliq}(\mathscr {H})=\{\mathscr {D}_1,\ldots ,\mathscr {D}_m\}$ be the cliques of $\mathscr {H}$ . To simplify the notation, assume that ${\mathcal L}(\Gamma )={\mathcal L}(\Lambda )={\mathcal M}$ . Then, by Theorem 7.1, for any $1\leq i\leq n$ , one can find a unitary $w_i\in {\mathcal M}$ and a subgroup $\Lambda _i<\Lambda $ satisfying the following relations:

1. 1. $\mathbb T w_i \Gamma _{\mathscr {C}_i} w_i^* =\mathbb T \Lambda _i$ for all $1\leq i\leq n$ ;

2. 2. The virtual centralizers satisfy $vC_\Lambda (\Lambda _i)= 1$ for all $1\leq i\leq n$ .

3. 3. $\mathbb T w_i \Gamma _{\mathscr {C}_{i,i+1}} w_i^*= \mathbb T w_{i+1} \Gamma _{\mathscr {C}_{i,i+1}} w_{i+1}^*= \mathbb T \Lambda _{i,i+1}$ where $\Lambda _{i,i+1}= \Lambda _i\cap \Lambda _{i+1}$ for all $1\leq i\leq n-1$ ;

4. 4. There is an $s\in \Lambda $ , such that $\mathbb T w_n \Gamma _{\mathscr {C}_{n,1}} w_n^*= \mathbb T v_s w_{1} \Gamma _{\mathscr {C}_{n,1}} w_{1}^*= \mathbb T \Lambda _{n}\cap s \Lambda _1 s^{-1}$ ;

Items 1., 3. and 4. are clear. We only have to justify 2. For seeing this, note that Lemma 2.15 and Theorem 2.16 give ${\mathcal L}(\mathrm {QN}_\Lambda (\Lambda _i))=\mathscr {QN}_{{\mathcal M}}({\mathcal L}(\Lambda _i))"= w_i \mathscr {QN}_{{\mathcal M}}({\mathcal L}(\Gamma _{\mathscr {C}_i}))" w_i^*= w_i {\mathcal L}(\Gamma _{\mathscr {C}_i}) w_i^*={\mathcal L}(\Lambda _i)$ . Since $vC_{\Lambda }(\Lambda _i) < \mathrm {QN}_\Lambda (\Lambda _i)$ , we deduce that $vC_{\Lambda }(\Lambda _i)<\Lambda _i$ , and hence, by using that $\Lambda _i$ is icc, we derive $vC_{\Lambda }(\Lambda _i)=vC_{\Lambda _i}(\Lambda _i)=1$ .

Claim 7.7. There exist a map $\sigma :\{1,\dots ,n\}\to \{1,\dots ,m\}$ and elements $h_1,\dots ,h_n \in \Lambda $ so that $\Lambda _i\leqslant h_i\Lambda _{\mathscr {D}_{\sigma (i)}} h^{-1}_i$ , for any $1\leq i\leq n$ .

Proof of Claim 7.7.

Let $1\leq i\leq n$ be arbitrary. Since $\Gamma _{\mathscr {C}_i}$ has property (T), then $\Lambda _i< \Lambda $ has property (T) as well. By applying Theorem 2.8, one can find $1\leq k\leq m$ , such that ${\mathcal L}(\Lambda _i)\prec _{{\mathcal M}} {\mathcal L}(\Lambda _{\mathscr {D}_k}).$ Using Lemma 2.6, there is $h\in \Lambda $ , such that $[\Lambda _i: \Lambda _i \cap h \Lambda _{\mathscr {D}_k} h^{-1}]<\infty $ . Thus, there is a finite index normal subgroup $\Lambda _i'\lhd \Lambda _i$ , such that $\Lambda ^{\prime }_i\leqslant h\Lambda _{\mathscr {D}_k} h^{-1}$ . We continue by proving that $\Lambda _i\leqslant h\Lambda _{\mathscr {D}_k} h^{-1}$ . To this end, fix $t\in \Lambda _i$ . Hence, $\Lambda _i'\leqslant h\Lambda _{\mathscr {D}_k} h^{-1} \cap th\Lambda _{\mathscr {D}_k} h^{-1}t^{-1}$ , and by Proposition 2.1, we have that

(7.4) $$ \begin{align}h\Lambda_{\mathscr{D}_k} h^{-1} \cap th\Lambda_{\mathscr{D}_k} h^{-1}t^{-1}= h_0 h\Lambda_{\mathscr{T}} h^{-1} h_0^{-1}\text{ for some }\mathscr{T}\subseteq \mathscr{D}_k \text{ and }h_0\in h\Lambda_{\mathscr{D}_k} h^{-1}. \end{align} $$

Since $[\Lambda _i:\Lambda _i']<\infty $ , condition 2. implies that $vC_{\Lambda }(\Lambda _i')=vC_{\Lambda }(\Lambda _i)=1$ . It follows from (7.4) that $vC_{\Lambda }(h_0 h\Lambda _{\mathscr {T}} h^{-1} h_0^{-1})=1$ , and hence, $\mathscr {T}$ is a clique. This forces that $\mathscr {T}=\mathscr {D}_k$ , and hence, (7.4) implies that $th\Lambda _{\mathscr {D}_k} h^{-1}t^{-1} \geqslant h\Lambda _{\mathscr {D}_k} h^{-1}$ . Using Proposition 2.1, we derive that $th\Lambda _{\mathscr {D}_k} h^{-1}t^{-1} = h\Lambda _{\mathscr {D}_k} h^{-1}$ , and hence, $h^{-1}t h\in \Lambda _{\mathscr {D}_k\cup \mathrm {link}(\mathscr {D}_k)}$ . Since $\mathscr {D}_k$ is a clique, we conclude that $t\in h\Lambda _{\mathscr {D}_k} h^{-1}$ .

We continue with the following claim.

Claim 7.8. For every $1\leq i\leq n$ , we have $[h_i\Lambda _{\mathscr {D}_{\sigma (i)}} h^{-1}_i: \Lambda _i]<\infty $ .

Proof of Claim 7.8.

Fix $1\leq i\leq n$ and $v\in \mathscr {C}_i^{\mathrm {int}}$ , and note that $\mathrm {star}(v)=\mathscr {C}_i$ . Let $j=\sigma (i)$ . Next, we briefly argue that $|\mathscr {D}_j|\geqslant 2$ . Assume by contradiction that $\mathscr {D}_j=\{s\}$ . As $\Lambda $ is a nontrivial graph product, there is a nontrivial group $\Sigma $ so that $\Lambda =\Lambda _s \ast \Sigma $ ; thus, ${\mathcal M} = {\mathcal L}(\Lambda _s\ast \Sigma )$ . Since ${\mathcal L}(\Gamma _{\mathscr {C}_{i,i+1}})$ is diffuse, then using Claim 7.7 and relation 1. we get $x {\mathcal L}(\Gamma _{\mathscr {C}_{i,i+1}}) x^*\subseteq \mathcal L(\Lambda _s)$ where we have denoted by $x:= u_{h_i^{-1}}w_i\in \mathscr {U}({\mathcal M})$ . Using Lemma 2.18 and the graph product structure of $\Gamma $ , this further implies that $x \mathcal L(\Gamma _{\mathscr {C}_{i}\cup \mathscr {C}_{i+1}}) x^*= \mathscr N_{{\mathcal M}} (x \mathcal L(\Gamma _{\mathscr {C}_{i,i+1}}) x^*)"\subseteq \mathcal L(\Lambda _s)$ . In particular, we have $ x {\mathcal L}(\Gamma _{\mathscr {C}_{i+1,i+2}}) x^*\subseteq \mathcal L(\Lambda _s)$ . Thus, taking the normalizer and repeating the previous argument, we get $ x \mathcal L(\Gamma _{\mathscr {C}_{i+1}\cup \mathscr {C}_{i+2}}) x^*= \mathscr N_{{\mathcal M}} (x \mathcal L(\Gamma _{\mathscr {C}_{i+1,i+2}}) x^*)"\subseteq {\mathcal L}(\Lambda _s)$ . Therefore, proceeding by induction, we have $ x {\mathcal L}(\Gamma _{\mathscr {C}_{k}\cup \mathscr {C}_{l}}) x^*\subseteq \mathcal L(\Lambda _s)$ for all $1\leqslant k,l\leqslant m$ satisfying $\hat k-\hat l \in \{ \hat 1,\widehat {n-1}\}$ ; here, the classes are considered in $\mathbb Z_m$ . Altogether, these relations imply that $ {\mathcal M} = x_i {\mathcal L}(\Gamma ) x^*= x \mathcal L(\vee _{k,l, \hat k-\hat l \in \{\hat 1,\widehat {n-1}\}} \Gamma _{\mathscr {C}_{k}\cup \mathscr {C}_{l}}) x^*=\bigvee _{k,l, \hat k-\hat l \in \{\hat 1,\widehat {n-1}\}}x {\mathcal L}(\Gamma _{\mathscr {C}_{k}\cup \mathscr {C}_{l}}) x^*\subseteq {\mathcal L}(\Lambda _s) $ , contradicting $\Sigma \neq 1$ .

Now, fix an arbitrary $w\in \mathscr {D}_j$ . By letting $\mathcal P_1= {\mathcal L}(\Lambda _{\mathscr {D}_{j} \setminus \{w\}})$ and $\mathcal P_2= {\mathcal L}(\Lambda _w)$ , we have $\mathcal P_1\bar \otimes \mathcal P_2={\mathcal L}(\Lambda _{\mathscr {D}_j})$ . Since the vertex groups of $\Lambda $ are infinite, then by [CdSS17, Theorem 3.1], one of the following must hold:

1. 1. $\mathcal P_1\vee \mathcal P_2$ is amenable relative to $ {\mathcal L}(\Gamma _{\mathrm {link}(v) })$ inside ${\mathcal M}$ ;

2. 2. $\mathcal P_1\vee \mathcal P_2 \prec _{\mathcal M} {\mathcal L}(\Gamma _{\mathscr {G}\setminus \{v\}})$ ;

3. 3. $\mathcal P_1\vee \mathcal P_2 \prec _{\mathcal M} {\mathcal L}(\Gamma _{\mathrm {star}(v)})$ ;

4. 4. $\mathcal P_i \prec _{\mathcal M} {\mathcal L}(\Gamma _{\mathrm {link}(v) })$ , for some $1 \leq i\leq 2$ ;

First, assume 1. holds. Since $\mathbb T h_i^{-1} w_i \Gamma _{\mathscr {C}_i} w_i^* h_i=\mathbb T h_i^{-1} \Lambda _i h_i\subset \mathcal P_1\vee \mathcal P_2$ , we derive that $h_i^{-1} w_i {\mathcal L}(\Gamma _{\mathscr {C}_i}) w_i^* h_i$ is amenable relative to $ {\mathcal L}(\Gamma _{\mathrm {link}(v) })$ inside ${\mathcal M}$ . Since ${\mathcal L}(\Gamma _{\mathscr {C}_i})$ has property (T), then $h_i^{-1} w_i {\mathcal L}(\Gamma _{\mathscr {C}_i}) w_i^* h_i\prec _{{\mathcal M}} {\mathcal L} (\Gamma _{\mathrm {link}(v)})$ . By applying Corollary 2.7, it follows that $\mathscr {C}_i\subset \mathrm {link}(v)$ , contradiction.

Now, assume 2. holds. Since $\mathbb T h_i^{-1} w_i \Gamma _{\mathscr {C}_i} w_i^* h_i=\mathbb T h_i^{-1} \Lambda _i h_i\subset \mathcal P_1\vee \mathcal P_2$ , we derive that ${\mathcal L}(\Gamma _{\mathscr {C}_i})\prec _{\mathcal M} {\mathcal L} (\Gamma _{\mathscr {G}\setminus \{v\}})$ . As in the previous case, we apply Corollary 2.7 and derive that $\mathscr {C}_i\subset \mathscr {G} \setminus \{v\}$ , contradiction.

Next, assume 3. holds. As $\mathrm {star}(v)=\mathscr {C}_i$ and $\mathbb T w_i \Gamma _{\mathscr {C}_i} w_i^* =\mathbb T\Lambda _i $ , we derive that ${\mathcal L} (\Lambda _{\mathscr {D}_j})\prec _{\mathcal M} {\mathcal L} (h_i^{-1} \Lambda _i h_i)$ . This implies that there exist projections $p\in {\mathcal L} (\Lambda _{\mathscr {D}_j}), q\in \mathcal L (h_i^{-1} \Lambda _i h_i)$ , a nonzero partial isometry $w\in q {\mathcal M} p$ and a $*$ -homomorphism $\theta : p\mathcal L (\Lambda _{\mathscr {D}_j})p\to q\mathcal L (h_i^{-1} \Lambda _i h_i)q$ , such that $\theta (x)w=wx$ . Since $\Lambda _i\leqslant h_i\Lambda _{\mathscr {D}_{j}} h^{-1}_i$ , it follows that $w p\mathcal L (\Lambda _{\mathscr {D}_j})p\subset q \mathcal L (\Lambda _{\mathscr {D}_j})qw$ . Since ${\mathcal D}_j$ is a clique, it follows from Lemma 2.18 that $w\in \mathcal L (\Lambda _{\mathscr {D}_j})$ . This shows that ${\mathcal L} (\Lambda _{\mathscr {D}_j})\prec _{{\mathcal L} (\Lambda _{\mathscr {D}_j})} {\mathcal L} (h_i^{-1} \Lambda _i h_i)$ , which further implies that $[\Lambda _{\mathscr {D}_j}: h_i^{-1} \Lambda _i h_i]<\infty $ . This proves the claim.

Before completing Claim 7.8 by assuming that 4. holds, we first establish the following notation and prove the following subclaim. Since $\mathbb T w_i \Gamma _{\mathscr {C}_i} w_i^*=\mathbb T \Lambda _i$ , we define the subgroups $\Lambda ^i_v, \Lambda ^i_{\hat v}<\Lambda _i$ by $\mathbb T w_i \Gamma _{v} w_i^*=\mathbb T \Lambda ^i_{ v}$ and $\mathbb T w_i \Gamma _{\mathrm {link}(v)} w_i^*=\mathbb T \Lambda ^i_{\hat v}$ , respectively. Note that $\Lambda _i= \Lambda ^i_{v}\times \Lambda ^i_{\hat v}$ since $v\in \mathscr {C}_i^{\mathrm {int}}$ .

Subclaim 1. For any subset $\mathscr {D}_0\subset \mathscr {D}_j$ satisfying ${\mathcal L}(\Lambda _{\mathscr {D}_0})\prec _{{\mathcal M}} {\mathcal L} (\Gamma _{\mathrm {link}(v)})$ , the subgroup $(\Lambda _0)_{\mathscr {D}_0}:= h_i \Lambda _{\mathscr {D}_0} h_i^{-1}\cap \Lambda ^i_{\hat v} $ satisfies $[h_i \Lambda _{\mathscr {D}_0} h_i^{-1}:(\Lambda _0)_{\mathscr {D}_0}]<\infty $ .

Proof of Subclaim 1.

For proving the subclaim, we consider a subset $\mathscr {D}_0\subset \mathscr {D}_j$ satisfying ${\mathcal L}(\Lambda _{\mathscr {D}_0})\prec _{{\mathcal M}} \mathcal L (\Gamma _{\mathrm {link}(v)})$ . By applying Lemma 2.6, there is $k\in \Lambda $ , such that

(7.5) $$ \begin{align} [h_i \Lambda_{\mathscr{D}_0} h_i^{-1}: h_i \Lambda_{\mathscr{D}_0} h_i^{-1}\cap k \Lambda^i_{\hat v} k^{-1}]<\infty. \end{align} $$

Set $\Lambda _0= k^{-1}h_i \Lambda _{\mathscr {D}_0} h_i^{-1}k\cap \Lambda ^i_{\hat v}$ . We continue by showing that k normalizes $h_i \Lambda _{\mathscr {D}_0} h_i^{-1}$ . Recall that $\Lambda ^i_{\hat v}<\Lambda _i< h_i \Lambda _{\mathscr {D}_j} h_i^{-1}$ . From (7.5), we get that $[h_i \Lambda _{\mathscr {D}_0} h_i^{-1}: k \Lambda _0 k^{-1}]<\infty $ , and consequently, there is a finite subset $S\subset \Lambda $ satisfying $(h_i \Lambda _{\mathscr {D}_0} h_i^{-1})k\subset \cup _{s\in S} sk \Lambda _0 \subset \cup _{s\in S} sk h_i \Lambda _{\mathscr {D}_j}h_i^{-1}$ . By applying Theorem 2.16 we derive that $h_i^{-1}k h_i\in \Lambda _{\mathscr {D}_j\cup \mathrm {link}(\mathscr {D}_0)}$ , which implies that k is normalizing $h_i \Lambda _{\mathscr {D}_0} h_i^{-1}$ . In combination with (7.5), this shows that ${\Lambda _0=(\Lambda _0)_{\mathscr {D}_0}}$ is a finite index subgroup of $h_i \Lambda _{\mathscr {D}_0} h_i^{-1}$ . This proves the subclaim.

Now, we can assume that 4. holds for all $w\in \mathscr {D}_j$ . Namely, one of the following holds: a) $\mathcal P_1 \prec _{\mathcal M} {\mathcal L}(\Gamma _{\mathrm {link}(v) })$ or b) $\mathcal P_2 \prec _{\mathcal M} {\mathcal L}(\Gamma _{\mathrm {link}(v) })$ . We continue by considering the following cases.

Case 1. Suppose there are $w_1\neq w_2\in \mathscr {D}_j$ , such that a) holds. The subclaim implies that $[h_i \Lambda _{\mathscr {D}_j\setminus \{w_1\}} h_i^{-1}:(\Lambda _0)_{\mathscr {D}_j\setminus \{w_1\}}]<\infty $ and $[h_i \Lambda _{\mathscr {D}_j\setminus \{w_2\}} h_i^{-1}:(\Lambda _0)_{\mathscr {D}_j\setminus \{w_2\}}]<\infty $ . Since ${w_1\in \mathscr {D}_j\setminus \{w_2\}}$ , it follows that $[h_i\Lambda _{w_1}h_i^{-1}: h_i\Lambda _{w_1}h_i^{-1}\cap \Lambda ^i_{\hat v}]<\infty $ . Consequently, the inclusion $(h_i\Lambda _{w_1}h_i^{-1}\cap \Lambda ^i_{\hat v})\vee (\Lambda _0)_{\mathscr {D}_j\setminus \{w_1\}}<\Lambda _i<h_i \mathscr {D}_j h_i^{-1}$ is of finite index. This proves the claim.

Case 2. Next, we suppose that $b)$ holds for any $w\in \mathscr {D}_j$ . Using the subclaim, we get that $(\Lambda _0)_{w}<h_i \Lambda _{w} h_i^{-1}$ has finite index for any $w\in \mathscr {D}_j$ . Hence, the inclusion $\vee _{w\in \mathscr {D}_j}(\Lambda _0)_{w}<\Lambda _i<\vee _{w\in \mathscr {D}_j}h_i \Lambda _{w} h_i^{-1}=h_i \Lambda _{\mathscr {D}_j} h_i ^{-1}$ has finite index as well, which again proves the claim.

Case 3. Finally, assume that there is $w_0\in \mathscr {D}_j$ , such that a) holds and for any ${w\in \mathscr {D}_j\setminus \{w_0\}}$ , we have that b) holds. We note in passing that in this situation, b) does not provide any additional information besides a). Note that Subclaim 1 provides

(7.6) $$ \begin{align} [h_i \Lambda_{\mathscr{D}_j\setminus\{w_0\}} h_i^{-1}:(\Lambda_0)_{\mathscr{D}_j\setminus\{w_0\}}]<\infty. \end{align} $$

We continue by showing the following.

Subclaim 2. The group $(\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}}$ is icc and $ vC_{h_i\Lambda _{\mathscr {D}_j \setminus \{w_0\}}h_i^{-1} }((\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}})=1$ . Also, there exists a subgroup $\Upsilon < h_i \Lambda _{w_0} h_i^{-1}$ satisfying $[\Lambda _i: (\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}}\times \Upsilon ]<\infty $ .

Proof of Subclaim 2.

Using $(\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}}<\Lambda ^i_{\hat v}<h_i \Lambda _{\mathscr {D}_j\setminus \{w_0\}} h_i^{-1}\times h_i \Lambda _{w_0} h_i^{-1}$ , we apply Lemma 7.4 and deduce that there is a subgroup $\Lambda '< h_i \Lambda _{w_0} h_i^{-1}\cap \Lambda _{\hat v}^i$ , such that

(7.7) $$ \begin{align} [\Lambda_{\hat v}^i: (\Lambda_0)_{\mathscr{D}_j\setminus\{w_0\}}\times \Lambda']<\infty. \end{align} $$

Since $\Lambda _{\hat v}^i$ is icc, (7.7) implies that $(\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}}$ is icc as well. Next, we argue that there is a finite index subgroup $\Lambda " \leqslant h_i \Lambda _{w_0} h_i^{-1}\cap \Lambda _v^i$ , such that

(7.8) $$ \begin{align} [\Lambda^i_v: \Lambda"]<\infty. \end{align} $$

Indeed, as $(\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}}$ is icc, one can see from (7.6) that $C_{h_i \Lambda _{\mathscr {D}_j\setminus \{w_0\}} h_i^{-1}}((\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}})=1$ , and thus, $\Lambda _v^i\leqslant C_{h_i\Lambda _{\mathscr {D}_j}h_i^{-1}}(\Lambda ^i_{\hat v} ) \leqslant C_{h_i \Lambda _{\mathscr {D}_j}h_i^{-1}}((\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}} ) \leqslant h_i \Lambda _{w_0}h_i^{-1}$ . This shows that we can actually choose $\Lambda "=\Lambda _v^i$ to satisfy (7.8).

Altogether, (7.7) and (7.8) imply the existence of $\Upsilon := \Lambda '\times \Lambda "< h_i \Lambda _{w_0} h_i^{-1}$ , such that $[\Lambda _i: (\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}}\times \Upsilon ]<\infty $ . Together with condition 2., we deduce that $1= vC_{\Lambda }(\Lambda _i)= vC_{\Lambda }((\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}}\times \Upsilon )\geq vC_{h_i\Lambda _{\mathscr {D}_j \setminus \{w_0\}}h_i^{-1} }((\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}})\times vC_{h_i\Lambda _{w_0}h_i^{-1} }( \Upsilon )$ . In particular, this shows that $ vC_{h_i\Lambda _{\mathscr {D}_j \setminus \{w_0\}}h_i^{-1} }((\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}})=1.$

We continue by applying Lemma 7.5 and find a subgraph $v\in \mathscr {C}^1_i\subset \mathscr {C}_i$ , a finite index subgroup $\Upsilon _0\leqslant \Upsilon $ and finite index subgroups $\Gamma ^{\prime }_w\leqslant \Gamma _w$ , for any $w\in \mathscr {C}^1_i$ , such that if $\Gamma ^{\prime }_{\mathscr {C}_i^1}:= \times _{w\in \mathscr {C}_i^1} \Gamma ^{\prime }_w$ , then

(7.9) $$ \begin{align}\mathbb T w_i \Gamma^{\prime}_{\mathscr{C}_i^1} w_i^*=\mathbb T \Upsilon_0. \end{align} $$

By Lemma 7.5, we consider some finite index subgroups $\Gamma ^{\prime }_{\mathscr {C}_i \setminus \mathscr {C}_i^1}\leqslant \Gamma _{\mathscr {C}_i\setminus \mathscr {C}_i^1}$ and $(\Lambda ^0_0)_{\mathscr {D}_j\setminus \{w_0\}}< (\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}}$ , such that

(7.10) $$ \begin{align} \mathbb T w_i \Gamma^{\prime}_{\mathscr{C}_i \setminus\mathscr{C}_i^1} w_i^*=\mathbb T (\Lambda^0_0)_{\mathscr{D}_j\setminus\{w_0\}}. \end{align} $$

Since $\Gamma ^{\prime }_{\mathscr {C}_i^1}\leqslant \Gamma _{\mathscr {C}_i^1}$ has finite index and $v\in \mathscr {C}_i^1\cap \mathscr {C}_i^{\mathrm {int}}$ , we obtain from Theorem 2.16 that ${\mathcal L}(\Gamma _{\mathscr {C}_i \setminus \mathscr {C}_i^1})= \mathcal L(\Gamma ^{\prime }_{\mathscr {C}_i^1})' \cap {\mathcal M}$ . Hence, by taking relative commutants in (7.9) and using $\Upsilon _0<\Upsilon < h_i\Lambda _{w_0}h_i^{-1}$ , we deduce that $w_i \mathcal L(\Gamma _{\mathscr {C}_i \setminus \mathscr {C}_i^1}) w_i^*= w_i \mathcal L(\Gamma ^{\prime }_{\mathscr {C}_i^1})' w_i^*\cap {\mathcal M} = \mathcal L(\Upsilon _0)'\cap {\mathcal M} \supseteq \mathcal L(h_i\Lambda _{\mathrm {link}(w_0)}h_i^{-1}) $ . Since $(\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}}\leqslant h_i \Lambda _{\mathscr {D}_j\setminus \{w_0\}} h_i^{-1}$ , then relation (7.10) entails $w_i \mathcal L(\Gamma ^{\prime }_{\mathscr {C}_i \setminus \mathscr {C}_i^1}) w_i^*\subseteq \mathcal L(h_i \Lambda _{\mathscr {D}_j }h_i^{-1})$ , and hence, by combining this with the prior containment, it follows that ${\mathcal L}(h_i\Lambda _{\mathrm { link}(w_0)}h_i^{-1})\prec _{{\mathcal M}} {\mathcal L}(h_i \Lambda _{\mathscr {D}_j }h_i^{-1})$ . Using Corollary 2.7, we further deduce that $\mathrm {link}(w_0)\subset \mathscr {D}_j$ .

Next, we claim that

(7.11) $$ \begin{align}\mathscr{C}^1_i\subseteq \mathscr{C}_i^{\mathrm{int}}.\end{align} $$

Assume, by contradiction, that (7.11) does not hold. Without loss of generality, we may assume there is $v_0\in \mathscr {C}^1_i\cap \mathscr {C}_{i+1}$ . Note that $\mathbb T w_i \Gamma ^{\prime }_{v_0} w_i^*= \mathbb T \Upsilon _1$ for some subgroup $\Upsilon _1< \Upsilon _0< h_i \Lambda _{w_o}h_i^{-1}$ , and hence, $ w_i{\mathcal L}(\Gamma ^{\prime }_{v_0}) w_i^*= {\mathcal L}(\Upsilon _1)$ . By taking relative commutants and using Corollary 2.9 and $\mathrm {link}(w_0)=\mathscr {D}_j\setminus \{w_0\}$ , we see that

(7.12) $$ \begin{align}\begin{aligned} \mathcal{Q}:=w_i{\mathcal L}(\Gamma_{\mathrm{link}(v_0) })w_i^*&= (w_i{\mathcal L}(\Gamma^{\prime}_{v_0}) w_i^*)'\cap {\mathcal M} = {\mathcal L}(\Upsilon_1)'\cap {\mathcal M}\\ &= {\mathcal L}(h_i\Lambda_{\mathscr{D}_j\setminus \{w_0\}} h_i^{-1})\bar\otimes ({\mathcal L}(\Upsilon_1)'\cap {\mathcal L}(h_i \Lambda_{w_0}h_i^{-1})).\end{aligned} \end{align} $$

Since $\mathrm {link}(v_0)=(\mathscr {C}_i\cup \mathscr {C}_{i+1})\setminus \{v_0\}$ , we note that

$$ \begin{align*}\mathcal{Q}=w_i{\mathcal L}(\Gamma_{\mathrm{link}(v_0) })w_i^*= w_i{\mathcal L}(\Gamma_{\mathscr{C}_{i,i+1}\setminus \{v_0\} })w_i^* \bar\otimes (w_i\mathcal L(\Gamma_{\mathscr{C}_i \setminus \mathscr{C}_{i,i+1}})w_i^* \ast w_i \mathcal L(\Gamma_{\mathscr{C}_{i+1} \setminus \mathscr{C}_{i,i+1}}) w_i^*).\end{align*} $$

Since $ {\mathcal L} (h_i\Lambda _{\mathscr D_j\setminus \{w_0\}} h_i^{-1})$ is a regular property (T) von Neumann subalgebra of $\mathcal {Q}$ , it follows from the main technical result of [IPP05] (see also [PV09, Theorem 5.4]) that ${\mathcal L} (h_i\Lambda _{\mathscr D_j\setminus \{w_0\}} h_i^{-1})\prec _{\mathcal {Q}} w_i{\mathcal L}(\Gamma _{\mathscr {C}_{i,i+1}\setminus \{v_0\} })w_i^*$ . Thus, relation (7.10) and Corollary 2.7 entail

(7.13) $$ \begin{align} \mathscr{C}_{i} \setminus \mathscr{C}^{1}_{i}\subseteq \mathscr{C}_{i,i+1}\setminus \{v_0\}. \end{align} $$

Using (7.10), we have $w_i {\mathcal L}(\Gamma ^{\prime }_{\mathscr {C}_{i} \setminus \mathscr {C}_{i}^{1}} ) w_i^*=\mathcal L((\Lambda _0^0)_{\mathscr {D}_j\setminus \{w_0\}})$ . By taking relative commutants in (7.12) inside $\mathcal {Q}$ and using Subclaim 2 and (7.13), we further get

(7.14) $$ \begin{align} w_i{\mathcal L}(\Gamma_{\mathscr{C}_{i}\cup \mathscr{C}_{i+1} \setminus(\mathscr{C}_{i}\setminus \mathscr {C}^{1}_{i} )})w_i^*={\mathcal L}(\Upsilon_1)'\cap {\mathcal L}(h_i \Lambda_{w_0}h_i^{-1})\subset {\mathcal L}(h_i \Lambda_{w_0}h_i^{-1}). \end{align} $$

In particular, we have

(7.15) $$ \begin{align} w_i{\mathcal L}(\Gamma_{\mathscr{C}_{i+1} \setminus \mathscr{C}_{i,i+1} })w_i^*\subset {\mathcal L}(h_i \Lambda_{w_0}h_i^{-1}).\end{align} $$

Since we also have $\mathbb T w_{i+1}\Gamma _{\mathscr {C}_{i+1} \setminus \mathscr {C}_{i,i+1} }w_{i+1}^*< \mathbb T h_{i+1}\Lambda _{\mathscr {D}_{\sigma (i+1)}} h_{i+1}^{-1} $ , we obtain that $w_0\in \mathscr {D}_{\sigma (i+1)}$ . Indeed, by [Va10a, Lemma 2.7], there is $\lambda \in \Lambda $ , such that ${\mathcal L}(\Gamma _{\mathscr {C}_{i+1} \setminus \mathscr {C}_{i,i+1} })\prec _{{\mathcal M}}^s \mathcal L (\lambda \Lambda _{w_0} \lambda ^{-1} \cap \Lambda _{\mathscr {D}_{\sigma (i+1)}} )$ , and hence, $\lambda \Lambda _{w_0} \lambda ^{-1} \cap \Lambda _{\mathscr {D}_{\sigma (i+1)}}$ is an infinite group. By Proposition 2.1, we get that $w_0\in \mathscr {D}_{\sigma (i+1)}$ . Now, using that $\mathrm {link}(w_0)\subset \mathscr {D}_j$ , we get $\sigma (i+1)=j$ . In particular, we have that $\mathbb T w_{i+1}\Gamma _{\mathscr {C}_{i+1} \setminus \mathscr {C}_{i,i+1} }w_{i+1}^*=\mathbb T \Lambda _{i+1}< \mathbb T h_{i+1}\Lambda _{\mathscr {D}_{j}} h_{i+1}^{-1}.$ This further implies that

(7.16) $$ \begin{align}\mathbb T h_i h^{-1}_{i+1}w_{i+1} w_i^*( w_i\Gamma_{\mathscr{C}_{i+1} \setminus\mathscr{C}_{i,i+1} } w_i^*) w_i w_{i+1}^* h_{i+1} h_i^{-1}< \mathbb T h_{i}\Lambda_{\mathscr{D}_{j}} h_{i}^{-1}. \end{align} $$

Using Theorem 2.16, (7.15) and (7.16) further imply that $x:=h_i h^{-1}_{i+1}w_{i+1} w_i^*\in \mathscr {U}({\mathcal L}(h_i \Lambda _{\mathscr {D}_j} h_i^{-1}))$ . Thus, using Lemma 7.3, (7.15) and (7.16) further entail the existence of a finite index subgroup $\Gamma ^{\prime \prime }_{\mathscr {C}_{i+1} \setminus \mathscr {C}_{i,i+1}}\leqslant \Gamma _{\mathscr {C}_{i+1} \setminus \mathscr {C}_{i,i+1}}$ , such that $\mathbb T x( w_i\Gamma ^{\prime \prime }_{\mathscr {C}_{i+1} \setminus \mathscr {C}_{i,i+1} } w_i^*) x^* < \mathbb T h_{i}\Lambda _{w_0} h_{i}^{-1}$ . In particular, there exist a finite index subgroup $\Gamma ^{\prime \prime }_{\mathscr {C}_{i+1,i+2}}\leqslant \Gamma _{\mathscr {C}_{i+1,i+2}}$ and a subgroup $\Lambda _2<h_i \Lambda _{w_0} h_i^{-1}$ , such that $\mathbb T x( w_i\Gamma ^{\prime \prime }_{\mathscr {C}_{i+1,i+2}} w_i^*) x^* =\mathbb T \Lambda _2$ . Taking relative commutants and using Corollary 2.9, we get

(7.17) $$ \begin{align}x{\mathcal L}(\Gamma_{\mathscr{C}_{i+1}\cup \mathscr{C}_{i+2} \setminus \mathscr{C}_{i+1,i+2}})x^*= \mathcal L(h_i \Lambda_{\mathscr{D}_j\setminus \{w_0\}} h_i^{-1})\bar \otimes( {\mathcal L}(\Lambda_2)'\cap {\mathcal L}(h_i\Lambda_{w_0}h_i^{-1})). \end{align} $$

However, we canonically have the free product decomposition

(7.18) $$ \begin{align} x{\mathcal L}(\Gamma_{\mathscr{C}_{i+1}\cup \mathscr{C}_{i+2} \setminus \mathscr {C}_{i+1,i+2}})x^*= x{\mathcal L}(\Gamma_{\mathscr{C}_{i+1}\setminus \mathscr{C}_{i+1,i+2}}) x^*\ast x\mathcal L(\Gamma_{\mathscr{C}_{i+2} \setminus \mathscr{C}_{i+1,i+2}})x^*. \end{align} $$

From (7.17), we derive that ${\mathcal L}(h_i \Lambda _{\mathscr {D}_j\setminus \{w_0\}} h_i^{-1})$ is a regular property (T) von Neumann subalgebra of $x{\mathcal L}(\Gamma _{\mathscr {C}_{i+1}\cup \mathscr {C}_{i+2} \setminus \mathscr {C}_{i+1,i+2}})x^*$ . Hence, we obtain a contradiction by applying [PV09, Theorem 5.4] (see also [IPP05]) to relation (7.18). Thus, relation (7.11) holds.

Using (7.11) and taking relative commutants in (7.10), we obtain that $w_i {\mathcal L}(\Gamma _{\mathscr {C}_i^1})w_i^*= {\mathcal L} (h_i \Lambda _{w_0}h_i^{-1})$ . Combining this with (7.9), we conclude that $[h_i \Lambda _{w_0} h_i^{-1}: \Upsilon _0]<\infty $ . Thus, $[h_i\Lambda _{\mathscr {D}_j} h_i^{-1}: (\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}}\times \Upsilon _0 ]<\infty $ . Since $ (\Lambda _0)_{\mathscr {D}_j\setminus \{w_0\}}\times \Upsilon _0 <\Lambda _i< h_i\Lambda _{\mathscr {D}_j} h_i^{-1}$ , we conclude the proof of our claim in case 3.

Next, we prove the following claim.

Claim 7.9. For any $1\leq i\leq n$ , there exists a unitary $y_i \in {\mathcal M}$ , such that $\mathbb T y_i \Gamma _{\mathscr {C}_i} y_i^*= \mathbb T \Lambda _{\mathscr {D}_{\sigma (i)}}$ .

Proof of Claim 7.9.

Fix $1\leq i\leq n$ . From Claim 7.8, we have $[h_i\Lambda _{\mathscr {D}_{\sigma (i)}} h^{-1}_i: \Lambda _i]<\infty $ . Recall also that $\mathbb T w_i \Gamma _{\mathscr {C}_i} w_i^* =\mathbb T \Lambda _i$ , which clearly implies that $w_i {\mathcal L}(\Gamma _{\mathscr {C}_i}) w_i^* = \mathcal L(\Lambda _i)$ . Taking quasi-normalizers of these algebras and using successively part 2. in Lemma 2.15, we have

$$ \begin{align*} \begin{aligned}w_i {\mathcal L}(\Gamma_{\mathscr{C}_i}) w_i^*&= w_i {\mathcal L}(\mathrm{QN}_\Gamma(\Gamma_{\mathscr{C}_i})) w_i^*= w_i \mathscr Q\mathscr N ({\mathcal L}(\Gamma_{\mathscr{C}_i}))" w_i^*\\&= \mathscr Q\mathscr N (w_i{\mathcal L}(\Gamma_{\mathscr{C}_i})w_i^*)"= \mathscr{QN}_{\mathcal M}({\mathcal L}(\Lambda_i))"= {\mathcal L}(\mathrm{QN}_\Lambda(\Lambda_i)) \\& ={\mathcal L}(\mathrm{QN}_\Lambda(h_i \Lambda_{\mathscr{D}_{\sigma(i)}}h_i^{-1}))= {\mathcal L}(\mathrm{QN}_\Lambda(h_i \Lambda_{\mathscr{D}_{\sigma(i)}}h_i^{-1}))={\mathcal L}(h_i \Lambda_{\mathscr{D}_{\sigma(i)}}h_i^{-1})\\&= v_{h_i}{\mathcal L}(\Lambda_{\mathscr{D}_{\sigma(i)}}) v_{h_i^{-1}}.\end{aligned}\end{align*} $$

Thus, the claim follows from the $W^*$ -superrigidity of $\Gamma _{\mathscr {C}_i}$ (see Corollary 5.3).

In particular, Claim 7.9 implies that $y_i{\mathcal L}(\Gamma _{\mathscr {C}_i})y_i^*= {\mathcal L}(\Lambda _{\mathscr {D}_{\sigma (i)}})$ . Then, using Theorem 2.3 and the same argument from the proof of [CDD22, Theorem 7.9], one can find a unitary $u\in {\mathcal M}$ , such that $u{\mathcal L}(\Gamma _{\mathscr {C}_i})u^*= \mathcal L(\Lambda _{\mathscr {D}_{\sigma (i)}})$ for all i. Thus, without any loss of generality, we can assume that ${\mathcal L}(\Gamma _{\mathscr {C}_i})= {\mathcal L}(\Lambda _{\mathscr {D}_{\sigma (i)}})=:{\mathcal M}_i$ for all i. Since $\Gamma _{\mathscr {C}_i}$ is $W^*$ -superrigid, we can find unitaries $x_i\in {\mathcal M}_i$ , such that

(7.19) $$ \begin{align}\mathbb T x_i \Gamma_{\mathscr{C}_i} x_i^*= \mathbb T \Lambda_{\mathscr{D}_{\sigma (i)}}\text{ for all }i.\end{align} $$

These relations already show that $\Lambda = \vee _i \Lambda _{\mathscr {D}_{\sigma (i)}}$ and also $\sigma : \mathrm {cliq}(\mathscr {G}) \rightarrow \mathrm {cliq}(\mathscr {H})$ are bijections. Moreover, one can see that the graph $\mathscr {H}\in \mathrm {CC}_1$ . This and relations (7.19) show that $\Lambda $ can be represented as a graph product $\Lambda _{\mathscr {H}'}$ , where the vertex groups $\Lambda _w$ are property (T) wreath-like products as in Theorem 5.2 and the underlying graph $\mathscr {H}'$ is isometric to $\mathscr {G}$ . Hence, the desired conclusion follows from [CDD22, Theorem 7.10].