A dichotomy for topological full groups

Abstract

Given a minimal action $\alpha $ of a countable group on the Cantor set, we show that the alternating full group $\mathsf {A}(\alpha )$ is non-amenable if and only if the topological full group $\mathsf {F}(\alpha )$ is $C^*$ -simple. This implies, for instance, that the Elek–Monod example of non-amenable topological full group coming from a Cantor minimal $\mathbb {Z}^2$ -system is $C^*$ -simple.

1 Introduction

Given an action $\alpha $ of a group on the Cantor set X, the topological full group of $\alpha $ , denoted by $\mathsf {F}(\alpha )$ , is the group of homeomorphisms on X which are locally given by $\alpha $ .

In [JM13], Juschenko and Monod showed that topological full groups of Cantor minimal $\mathbb {Z}$ -systems are amenable. Together with results of Matui [Mat06], this gave rise to the first examples of infinite, simple, finitely generated, amenable groups. On the other hand, in [EM13], Elek and Monod constructed an example of a free minimal $\mathbb {Z}^2$ -subshift whose topological full group contains a free group.

A group $\Gamma $ is said to have the unique trace property if its reduced $C^*$ -algebra $C^*_r(\Gamma )$ has a unique tracial state and to be $C^*$ -simple if $C^*_r(\Gamma )$ is simple. In [BKKO17], Breuillard et al. showed that $\Gamma $ has the unique trace property if and only if it does not contain any non-trivial amenable normal subgroup, and in [Ken20], Kennedy showed that $\Gamma $ is $C^*$ -simple if and only if it does not contain any nontrivial amenable uniformly recurrent subgroup (URS).

By using this new characterization of $C^*$ -simplicity, Le Boudec and Matte Bon showed in [LBMB18] that the topological full group of a free minimal action of a countable non-amenable group on the Cantor set is $C^*$ -simple, and asked whether the same conclusion holds if one does not assume freeness. In [BS19], Brix and the author showed that it suffices to assume that the action is topologically free. In [KTD19], Kerr and Tucker-Drob obtained examples of $C^*$ -simple topological full groups coming from actions of amenable groups.

Given an action $\alpha $ of a group on the Cantor set, Nekrashevych introduced in [Nek19] the alternating full group of the action, which we denote by $\mathsf {A}(\alpha )$ . This is a normal subgroup of $\mathsf {F}(\alpha )$ generated by certain copies of finite alternating groups. It was shown in [Nek19] that if $\alpha $ is minimal, then $\mathsf {A}(\alpha )$ is simple and is contained in every nontrivial normal subgroup of $\mathsf {F}(\alpha )$ .

In [MB18], Matte Bon obtained a classification of URSs of topological full groups. By using this result, we show the following theorem.

Theorem (Theorem 3.5)

Let $\alpha $ be a minimal action of a countable group on the Cantor set. The following conditions are equivalent:

1. (i) $\mathsf {A}(\alpha )$ is non-amenable.

2. (ii) Any group H such that $\mathsf {A}(\alpha )\leq H\leq \mathsf {F}(\alpha )$ is $C^*$ -simple.

3. (iii) There exists a $C^*$ -simple group H such that $\mathsf {A}(\alpha )\leq H\leq \mathsf {F}(\alpha )$ .

As a consequence, we obtain the following corollary.

Corollary (Corollary 3.6)

Let $\alpha $ be a minimal action of a countable group on the Cantor set. Then $\mathsf {F}(\alpha )$ has the unique trace property if and only if it is $C^*$ -simple. If $\mathsf {A}(\alpha )=\mathsf {F}(\alpha )'$ , then $\mathsf {F}(\alpha )$ is non-amenable if and only if it is $C^*$ -simple.

It is still an open problem whether $\mathsf {A}(\alpha )$ always coincides with $\mathsf {F}(\alpha )'$ , but in many cases, this is known to be true. For example, it follows from results of Matui [Mat15] that this is the case for free Cantor minimal $\mathbb {Z}^n$ -systems. This implies that the example of non-amenable topological full group coming from an action of $\mathbb {Z}^2$ in [EM13] is $C^*$ -simple.

2 Preliminaries

2.1 Topological dynamics

Given a locally compact Hausdorff space X, we denote by $\mathcal {B}(X)$ the Borel $\sigma $ -algebra of X, and by $\mathcal {P}(X)$ the space of regular probability measures on X.

If $\Gamma $ is a group acting by homeomorphisms on X, we say that X is a locally compact $\Gamma $ -space. If X admits no nontrivial $\Gamma $ -invariant closed subspaces, then we say that X (or the action) is minimal. Given $U\subset X$ , let $\operatorname {St}_{\Gamma} (U)$ consist of the elements of $\Gamma $ which fix pointwise U, and $\operatorname {St}_{\Gamma} (U)^0$ consist of the elements of $\Gamma $ which fix pointwise a neighborhood of U. To ease the notation, given $x\in X$ , we let $\Gamma _x:=\operatorname {St}_{\Gamma} (\{x\})$ and $\Gamma _x^0:=\operatorname {St}_{\Gamma} (\{x\})^0$ .

Denote by $\operatorname {Sub}(\Gamma )$ the set of subgroups of $\Gamma $ , endowed with the Chabauty topology; this is the restriction to $\operatorname {Sub}(\Gamma )$ of the product topology on $\{0, 1\}^{\Gamma} $ , where every subgroup $\Lambda \in \operatorname {Sub}(G)$ is identified with its characteristic function $\boldsymbol {1}_\Lambda \in \{0, 1\}^{\Gamma} $ . Notice that the space of amenable subgroups $\operatorname {Sub}_{am}(\Gamma )$ is closed in $\operatorname {Sub}(\Gamma )$ . We consider $\operatorname {Sub}(\Gamma )$ as a compact $\Gamma $ -space under the action by conjugation. A subgroup $\Lambda \leq \Gamma $ is said to be confined if $\{e\}$ is not in the closure of the $\Gamma $ -orbit of $\Lambda $ .

An invariant random subgroup (IRS) is a $\Gamma $ -invariant regular probability measure on $\operatorname {Sub}(\Gamma )$ . We say an IRS is amenable if its support is contained in $\operatorname {Sub}_{am}(\Gamma )$ . By [BKKO17, Corollary 4.3] and [BDL16, Corollary 1.5], $\Gamma $ has the unique trace property if and only if its unique amenable normal subgroup is $\{e\}$ , if and only if its unique amenable IRS is $\delta _{\{e\}}$ .

A URS is a $\Gamma $ -invariant closed minimal subspace $\mathcal {U}\subset \operatorname {Sub}(\Gamma )$ . We say $\mathcal {U}$ is amenable if every element of $\mathcal {U}$ is amenable. By [Ken20, Theorem 4.1], $\Gamma $ is $C^*$ -simple if and only if its only amenable URS is $\{\{e\}\}$ . Alternatively, $\Gamma $ is $C^*$ -simple if and only if it does not contain any confined amenable subgroup.

Suppose $\Gamma $ is countable and X is a minimal compact $\Gamma $ -space. Let $\operatorname {Stab}_{\Gamma} \colon X\to \operatorname {Sub}(\Gamma )$ be the map given by $\operatorname {Stab}_{\Gamma} (x):=\Gamma _x$ and $\operatorname {Stab}_{\Gamma} ^0\colon X\to \operatorname {Sub}(\Gamma )$ be the map given by $\operatorname {Stab}_{\Gamma} ^0(x):=\Gamma _x^0$ , for $x\in X$ . Notice that $\operatorname {Stab}_{\Gamma} $ and $\operatorname {Stab}_{\Gamma} ^0$ are Borel measurable and $\Gamma $ -equivariant. Moreover, the set $Y:=\{x\in X:\Gamma _x=\Gamma _x^0\}$ is dense in X and $\operatorname {Stab}_{\Gamma} (Y)$ is a URS, the so-called stabilizer URS of the action $\Gamma \!\curvearrowright \! X$ (for a proof of these last claims, see [LBMB18, Section 2]).

2.2 Topological full groups

Fix an action $\alpha $ of a group $\Gamma $ on the Cantor set X. We say that a homeomorphism $h\colon U\to V$ between clopen subsets $U,V\subset X$ is locally given by $\alpha $ if there exist $g_1,\dots ,g_n\in \Gamma $ and clopen sets $A_1,\dots ,A_n\subset U$ such that $U=\bigsqcup _{i=1}^n A_i$ and $h|_{A_i}=g_i|_{A_i}$ , for $1\leq i \leq n$ . The topological full group of $\alpha $ , denoted by $\mathsf {F}(\alpha )$ , is the group of homeomorphisms $h\colon X\to X$ which are locally given by $\alpha $ .

Given $d\in \mathbb {N}$ , a d-multisection is a collection of d disjoint clopen sets $(A_i)_{i=1}^d\subset X$ and $d^2$ homeomorphisms $(h_{i,j}\colon A_i\to A_j)_{i,j=1}^d$ which are locally given by $\alpha $ and such that, for $1\leq i,j,k\leq d$ , it holds that $h_{j,k}h_{i,j}=h_{i,k}$ and $h_{i,i}=\mathrm {Id}_{A_i}$ .

Given $d\in \mathbb {N}$ , let $S_d$ and $A_d$ be the symmetric and alternating groups, respectively. Given a d-multisection $\mathcal {F}=((A_i)_{i=1}^d,(h_{i,j})_{i,j=1}^d)$ and $\sigma \in S_d$ , let $\mathcal {F}(\sigma )\in \mathsf {F}(\alpha )$ be given by $\mathcal {F}(\sigma )|_{A_i}:=h_{\sigma (i),i}$ , for $1\leq i \leq n$ and $\mathcal {F}(\sigma )(x)=x$ for $x\notin \bigsqcup _{i=1}^n A_i$ . The alternating full group $\mathsf {A}(\alpha )$ is the subgroup of $\mathsf {F}(\alpha )$ generated by

$$ \begin{align*}\{\mathcal{F}(\sigma):d\in\mathbb{N}, \mathcal{F} \text{ is a }d\text{-multisection, } \sigma\in A_d\}.\end{align*} $$

Notice that $\mathsf {A}(\alpha )$ is normal in $\mathsf {F}(\alpha )$ and that $\mathsf {A}(\alpha )$ is contained in the derived subgroup $\mathsf {F}(\alpha )'$ . If $\alpha $ is a minimal action of a countable group on the Cantor set, then $\mathsf {A}(\alpha )$ is simple [Nek19, Theorem 4.1].

Remark 2.1 Alternatively, $\mathsf {F}(\alpha )$ and $\mathsf {A}(\alpha )$ can be described as groups of bisections of the groupoid of germs of $\alpha $ . This is the point of view adopted in [MB18, Nek19]. Conversely, given an effective groupoid G with unit space $G^{(0)}$ homeomorphic to the Cantor set, denote by $\alpha $ the natural action of the topological full group of G on $G^{(0)}$ . Then the topological and alternating full groups of G coincide with $\mathsf {F}(\alpha )$ and $\mathsf {A}(\alpha )$ , respectively [NO19, Corollary 4.7].

3 $\boldsymbol{C}^{\boldsymbol{*}}$ -simplicity of full groups

Given a locally compact $\Gamma $ -space X and $U\subset X$ open not necessarily invariant, let

$$ \begin{align*}\mathcal{P}_{\Gamma}(U):=\{\mu\in\mathcal{P}(U):\forall g\in\Gamma\ \forall A\in\mathcal{B}(U),\ \ \mu(A\cap g^{-1}U)=\mu(gA\cap U)\}. \end{align*} $$

Alternatively, one can characterize $\mathcal {P}_{\Gamma} (U)$ as the measures $\mu \in \mathcal {P}(U)$ such that $\mu (gA)=\mu (A)$ for every $g\in \Gamma $ and $A\in \mathcal {B}(U)$ such that $gA\subset U$ .

Proposition 3.1 Let X be a compact $\Gamma $ -space and $U\subset X$ open such that $X=\Gamma. U$ . Then the map $j\colon \mathcal {P}_{\Gamma} (X)\to \mathcal {P}_{\Gamma} (U)$ given by $j(\nu ):=\frac {\nu |_{\mathcal {B}(U)} }{\nu (U)}$ is a well-defined bijection.

Proof Take $g_1,\dots ,g_n\in \Gamma $ such that $X=\bigcup _{i=1}^n g_iU$ . For $1\leq i \leq n$ , let

$$ \begin{align*}A_i:=U\setminus\bigcup_{j=1}^{i-1}g_i^{-1}g_j U.\end{align*} $$

Then $X=\bigsqcup _{i=1}^n g_iA_i$ . Given $\nu \in \mathcal {P}_{\Gamma} (X)$ , obviously $\nu (U)\geq 1/n$ , so that j is a well-defined map. Moreover, given $A\in \mathcal {B}(X)$ , we have

(1) $$ \begin{align} \nu(A)=\sum_{i=1}^n\nu(g_i A_i\cap A)=\sum_{i=1}^n\nu(A_i\cap g_i^{-1} A). \end{align} $$

Since each $A_i$ is contained in U, this implies that $\nu $ is determined by its restriction to $\mathcal {B}(U)$ .

If $\nu _1,\nu _2\in \mathcal {P}_{\Gamma} (X)$ are such that $j(\nu _1)=j(\nu _2)$ , then $\nu _1|_U=\frac {\nu _1(U)}{\nu _2(U)}\nu _2|_U$ . Furthermore, by (1), we have

$$ \begin{align*}1=\nu_1(X)=\sum_{i=1}^n\nu_1(A_i)=\sum_{i=1}^n\frac{\nu_1(U)}{\nu_2(U)}\nu_2(A_i)=\frac{\nu_1(U)}{\nu_2(U)}\nu_2(X)=\frac{\nu_1(U)}{\nu_2(U)},\end{align*} $$

and hence $\nu _1(U)=\nu _2(U)$ . Consequently, $\nu _1=\nu _2$ and j is injective.

Let us now show that j is surjective. Given $\mu \in \mathcal {P}_{\Gamma} (U)$ and $A\in \mathcal {B}(X)$ , let

$$ \begin{align*}\nu(A):=\sum_{i=1}^n\mu(A_i\cap g_i^{-1}A).\end{align*} $$

Given $B\in \mathcal {B}(U)$ , we have

$$ \begin{align*}\nu(B)=\sum_{i=1}^n\mu(A_i\cap g_i^{-1}B)=\sum_{i=1}^n\mu(g_iA_i\cap B)=\mu(B),\end{align*} $$

so that $\nu |_{\mathcal {B}(U)}=\mu $ .

We claim that $\nu $ is $\Gamma $ -invariant. Fix $A\in \mathcal {B}(X)$ and $g\in \Gamma $ , and we will show that $\nu (A)=\nu (gA)$ .

For $1\leq i\leq n$ , let $h_i:=g^{-1}g_i$ , $B_i:=A_i\cap g_i^{-1}A$ , and $C_i:=A_i\cap g_i^{-1}gA=A_i\cap h_i^{-1} A$ . By definition of $\nu $ , we have $\nu (A)=\sum _{i=1}^n\mu (B_i)$ and $\nu (gA)=\sum _{i=1}^n\mu (C_i)$ .

Moreover, one can readily check that $A=\bigsqcup _{i=1}^n g_i B_i=\bigsqcup _{i=1}^n h_iC_i$ .

For $1\leq i,j\leq n$ , let $B_{i,j}:=B_i\cap g_i^{-1}h_jC_j$ and $C_{i,j}:=h_j^{-1}g_iB_i\cap C_j$ .

Notice that, for $1\leq i\leq n$ ,

$$ \begin{align*}\bigsqcup_{j=1}^n B_{i,j}=B_i\cap g_i^{-1}A=B_i\end{align*} $$

and, for $1\leq j \leq n$ ,

$$ \begin{align*}\bigsqcup_{i=1}^n C_{i,j}=h_j^{-1}A\cap C_j=C_j.\end{align*} $$

Furthermore, $g_i B_{i,j}=h_jC_{i,j}$ , and hence $\mu (B_{i,j})=\mu (C_{i,j})$ , since $B_{i,j}$ and $C_{i,j}$ are contained in U for every $i,j$ . Therefore,

$$ \begin{align*} \nu(A)=\sum_{i=1}^n\mu(B_i)=\sum_{i,j=1}^n \mu(B_{i,j})=\sum_{i,j=1}^n\mu(C_{i,j})=\sum_{j=1}^n\mu(C_j)=\nu(gA). \end{align*} $$

Finally, we have that $j(\frac {\nu }{\nu (X)})=\frac {\nu |_{\mathcal {B}(U)}/\nu (X)}{\nu (U)/\nu (X)}=\nu |_{\mathcal {B}(U)}=\mu .$

Remark 3.2 Let $\Gamma \!\curvearrowright \! X$ and $\Lambda \!\curvearrowright \! Y$ be actions on compact spaces. The actions are said to be Kakutani equivalent [Li18, Definition 2.14] if there exist clopen sets $A\subset X$ and $B\subset Y$ such that $X=\Gamma .A$ , $Y=\Lambda. B$ , and the partial transformation groupoids obtained by restriction to A and B are isomorphic. Proposition 3.1 implies that Kakutani equivalence induces a bijection between $\mathcal {P}_{\Gamma} (X)$ and $\mathcal {P}_\Lambda (X)$ .

The proof of the following result is analogous to [NO19, Lemma 4.9(2)].

Lemma 3.3 Let $\alpha $ be a minimal action of a group $\Gamma $ on the Cantor set X. Given $U\subset X$ clopen, $x\in U$ , and $g\in \Gamma $ such that $g(x)\in U$ , there exists a neighborhood V of x and $h\in \operatorname {St}_{\mathsf {A}(\alpha )}(U^{\mathsf {c}})$ such that $g|_V=h|_V$ .

Proof Case 1: $g(x)\neq x$ . Take $k\in \Gamma $ such that $k(g(x))\in U\setminus \{x,g(x)\}$ . Let V be a clopen neighborhood of x such that V, $g(V)$ , and $kg(V)$ are disjoint subsets of U. Then the homeomorphisms $h_{2,1}:=g|_V$ and $h_{3,1}:=kg|_V$ give rise to a $3$ -multisection $\mathcal {F}$ such that $\mathcal {F}((123))|_V=g|_V$ and $\mathcal {F}((123))\in \operatorname {St}_{\mathsf {A}(\alpha )}(U^{\mathsf {c}})$ .

Case 2: $g(x)=x$ . Take $k\in \Gamma $ such that $k(x)\in U\setminus \{ x\}$ . By Case 1, there are $h_1,h_2\in \operatorname {St}_{\mathsf {A}(\alpha )}(U^{\mathsf {c}})$ and $V_1,V_2$ neighborhoods of x and $k(x)$ , respectively, such that $k|_{V_1}=h_1|_{V_1}$ and $gk^{-1}|_{V_2}=h_2|_{V_2}$ . Then $V:=V_1\cap k^{-1}(V_2)$ is a neighborhood of x such that $h_2h_1|_V=g|_V$ .

The next lemma uses the same idea of [MB18, Corollary 6.5].

Lemma 3.4 Let $\alpha $ be a minimal action of a countable group $\Gamma $ on the Cantor set X and H a group such that $\mathsf {A}(\alpha )\leq H\leq \mathsf {F}(\alpha )$ . Then H is not $C^*$ -simple if and only if $\mathsf {A}(\alpha )_x^0$ is amenable for all $x\in X$ .

Proof Suppose H is not $C^*$ -simple. Then H contains a confined amenable subgroup. By [MB18, Theorem 6.1], there exists $Q\subset X$ finite such that $\operatorname {St}_{\mathsf {A}(\alpha )}^0(Q)$ is amenable. Given $x\in X$ , take a net $(g_i)\subset \mathsf {F}(\alpha )$ such that $g_iq\to x$ for any $q\in Q$ (existence of such a net $(g_i)$ follows from minimality and proximality of $\mathsf {F}(\alpha )\!\curvearrowright \! X$ ; see [MB18, Lemma 5.12]). Take K a limit point of $g_i\operatorname {St}_{\mathsf {A}(\alpha )}^0(Q)g_i^{-1}$ . One can readily check that $\mathsf {A}(\alpha )_x^0\leq K$ , and hence $\mathsf {A}(\alpha )_x^0$ is amenable.

Conversely, if $\mathsf {A}(\alpha )_x^0$ is amenable for all $x\in X$ , then since $\mathsf {A}(\alpha )_x^0$ is nontrivial for every x, it follows that the stabilizer URS $\mathcal {U}$ of $\mathsf {A}(\alpha )\!\curvearrowright \! X$ is a nontrivial amenable URS of $\mathsf {A}(\alpha )$ . By [MB18, Theorem 6.1], any element of $\mathcal {U}$ is a confined subgroup of $\mathsf {F}(\alpha )$ (hence of H as well). Therefore, H is not $C^*$ -simple.

Theorem 3.5 Let $\alpha $ be a minimal action of a countable group $\Gamma $ on the Cantor set X. The following conditions are equivalent:

1. (i) $\mathsf {A}(\alpha )$ is non-amenable.

2. (ii) Any group H such that $\mathsf {A}(\alpha )\leq H\leq \mathsf {F}(\alpha )$ is $C^*$ -simple.

3. (iii) There exists a $C^*$ -simple group H such that $\mathsf {A}(\alpha )\leq H\leq \mathsf {F}(\alpha )$ .

Proof The implications (ii) $\implies $ (iii) $\implies $ (i) are immediate.

(i) $\implies $ (ii): Suppose that there exists H non- $C^*$ -simple such that $\mathsf {A}(\alpha )\leq H\leq \mathsf {F}(\alpha )$ . By Lemma 3.4, $\mathsf {A}(\alpha )_x^0$ is amenable for every $x\in X$ .

Fix a clopen nonempty set U properly contained in X. Since, for any $x\in U^{\mathrm {c}}$ , we have $\Lambda :=\operatorname {St}_{\mathsf {A}(\alpha )}(U^{\mathsf {c}})\leq \mathsf {A}(\alpha )_x^0$ , it follows that $\Lambda $ is amenable.

Let $\mu \in \mathcal {P}_\Lambda (U)$ , and we claim that $\mu \in \mathcal {P}_{\Gamma} (U)$ . By regularity, it suffices to show that, for any $K\subset U$ compact and $g\in \Gamma $ such that $g(K)\subset U$ , it holds that $\mu (gK)=gK$ . By Lemma 3.3, there are $h_1,\dots ,h_n\in \Lambda $ and a partition $K=\bigsqcup _{i=1}^n K_i$ into compact sets such that $g|_{K_i}=h_i|_{K_i}$ for $1\leq i\leq n$ . Therefore,

$$ \begin{align*}\mu(gK)=\sum_{i=1}^n\mu(gK_i)=\sum_{i=1}^n\mu(h_iK_i)=\sum_{i=1}^n\mu(K_i)=\mu(K)\end{align*} $$

and $\mu \in \mathcal {P}_{\Gamma} (U)$ .

We conclude from Proposition 3.1 that there is $\nu \in \mathcal {P}_{\Gamma} (X)=\mathcal {P}_{\mathsf {F}(\alpha )}(X)$ . Furthermore, by minimality of the action, $\nu $ has full support. Let $\rho :=(\operatorname {Stab}_{\mathsf {A}(\alpha )}^0)_*\nu $ .

Given $g\in \Lambda \setminus \{e\}$ , we have $\rho (\{K\in \operatorname {Sub}(\mathsf {A}(\alpha )):g\in K\})\geq \nu (U^{\mathsf {c}})>0$ . Hence, $\rho $ is a nontrivial amenable IRS on $\mathsf {A}(\alpha )$ . Since $\mathsf {A}(\alpha )$ is simple, this implies that $\mathsf {A}(\alpha )$ is amenable.

The following is an immediate consequence of Theorem 3.5.

Corollary 3.6 Let $\alpha $ be a minimal action of a countable group on the Cantor set. Then $\mathsf {F}(\alpha )$ has the unique trace property if and only if it is $C^*$ -simple. If $\mathsf {A}(\alpha )=\mathsf {F}(\alpha )'$ , then $\mathsf {F}(\alpha )$ is non-amenable if and only if it is $C^*$ -simple.

Remark 3.7 If $\alpha $ is an action of a group on the noncompact Cantor set X, then the topological full group $\mathsf {F}(\alpha )$ is the group of homeomorphisms on X which are locally given by $\alpha $ and have compact support. Moreover, $\mathsf {A}(\alpha )$ is defined by requiring that the domains of the partial homeomorphisms of the multisections to be compact-open (as in [MB18, Definition 5.1]). By arguing as in [MB18, Corollary 6.5], the same conclusion of Theorem 3.5 and Corollary 3.6 holds in the noncompact case.

Example 3.8 It follows from [Mat12, Lemma 6.3] and [Mat15, Theorem 4.7] that, given a free minimal action $\alpha $ of $\mathbb {Z}^n$ on the Cantor set, it holds that $\mathsf {F}(\alpha )'=\mathsf {A}(\alpha )$ . Hence, the example of non-amenable topological full group coming from a Cantor minimal $\mathbb {Z}^2$ -system in [EM13] is $C^*$ -simple.