Minimal subdynamics and minimal flows without characteristic measures

Abstract

Given a countable group G and a G-flow X, a probability measure $\mu $ on X is called characteristic if it is $\mathrm {Aut}(X, G)$-invariant. Frisch and Tamuz asked about the existence of a minimal G-flow, for any group G, which does not admit a characteristic measure. We construct for every countable group G such a minimal flow. Along the way, we are motivated to consider a family of questions we refer to as minimal subdynamics: Given a countable group G and a collection of infinite subgroups $\{\Delta _i: i\in I\}$, when is there a faithful G-flow for which every $\Delta _i$ acts minimally?

Given a countable group G and a faithful G-flow X, we write $\mathrm {Aut}(X, G)$ for the group of homeomorphisms of X which commute with the G-action. When G is abelian, $\mathrm {Aut}(X, G)$ contains a natural copy of G resulting from the G-action, but in general this need not be the case. Much is unknown about how the properties of X restrict the complexity of $\mathrm {Aut}(X, G)$ ; for instance, Cyr and Kra [1] conjecture that when $G = \mathbb {Z}$ and $X\subseteq 2^{\mathbb {Z}}$ is a minimal, $0$ -entropy subshift, then $\mathrm {Aut}(X, \mathbb {Z})$ must be amenable. In fact, no counterexample is known even when restricting to any two of the three properties ‘minimal’, ‘ $0$ -entropy’ or ‘subshift’. In an effort to shed light on this question, Frisch and Tamuz [3] define a probability measure $\mu $ on X to be characteristic if it is $\mathrm {Aut}(X, G)$ -invariant. They show that $0$ -entropy subshifts always admit characteristic measures. More recently, Cyr and Kra [2] provide several examples of flows which admit characteristic measures for nontrivial reasons, even in cases where $\mathrm {Aut}(X, G)$ is nonamenable. Frisch and Tamuz asked (Question 1.5, [3]) whether there exists, for any countable group G, some minimal G-flow without a characteristic measure. We give a strong affirmative answer.

Theorem 0.1. For any countably infinite group G, there is a free minimal G-flow X so that X does not admit a characteristic measure. More precisely, there is a free $(G\times F_2)$ -flow X which is minimal as a G-flow and with no $F_2$ -invariant measure.

We remark that the X we construct will not in general be a subshift.

Over the course of proving Theorem 0.1, there are two main difficulties to overcome. The first difficulty is a collection of dynamical problems we refer to as minimal subdynamics. The general template of these questions is as follows.

Question 0.2. Given a countably infinite group $\Gamma $ and a collection $\{\Delta _i: i\in I\}$ of infinite subgroups of $\Gamma $ , when is there a faithful (or essentially free, or free) minimal $\Gamma $ -flow for which the action of each $\Delta _i$ is also minimal? Is there a natural space of actions in which such flows are generic?

In [8], the author showed that this was possible in the case $\Gamma = G\times H$ and $\Delta = G$ for any countably infinite groups G and H. We manage to strengthen this result considerably.

Theorem 0.3. For any countably infinite group $\Gamma $ and any collection $\{\Delta _n: n\in \mathbb {N}\}$ of infinite normal subgroups of $\Gamma $ , there is a free $\Gamma $ -flow which is minimal as a $\Delta _n$ -flow for every $n\in \mathbb {N}$ .

In fact, what we show when proving Theorem 0.3 is considerably stronger. Recall that given a countably infinite group $\Gamma $ , a subshift $X\subseteq 2^\Gamma $ is strongly irreducible if there is some finite symmetric $D\subseteq \Gamma $ so that whenever $S_0, S_1\subseteq \Gamma $ satisfy $DS_0\cap S_1 = \emptyset $ (i.e., $S_0$ and $S_1$ are D-apart), then for any $x_0, x_1\in X$ , there is $y\in X$ with $y|_{S_i} = x_i|_{S_i}$ for each $i< 2$ . Write $\mathcal {S}$ for the set of strongly irreducible subshifts, and write $\overline {\mathcal {S}}$ for its Vietoris closure. Frisch, Tamuz and Vahidi-Ferdowsi [5] show that in $\overline {\mathcal {S}}$ , the minimal subshifts form a dense $G_\delta $ subset. In our proof of Theorem 0.3, we show that the shifts in $\overline {\mathcal {S}}$ which are $\Delta _n$ -minimal for each $n\in \mathbb {N}$ also form a dense $G_\delta $ subset.

This brings us to the second main difficulty in the proof of Theorem 0.1. Using this stronger form of Theorem 0.3, one could easily prove Theorem 0.1 by finding a strongly irreducible $F_2$ -subshift which does not admit an invariant measure. This would imply the existence of a strongly irreducible $(G\times F_2)$ -subshift without an $F_2$ -invariant measure. As not admitting an $F_2$ -invariant measure is a Vietoris-open condition, the genericity of G-minimal subshifts would then be enough to obtain the desired result. Unfortunately, whether such a strongly irreducible subshift can exist (for any nonamenable group) is an open question. To overcome this, we introduce a flexible weakening of the notion of a strongly irreducible shift.

The paper is organized as follows. Section $1$ is a very brief background section on subsets of groups, subshifts and strong irreducibility. Section $2$ introduces the notion of a UFO, a useful combinatorial gadget for constructing shifts where subgroups act minimally; Theorem 0.3 answers Question 3.6 from [8]. Section $3$ introduces the notion of $\mathcal {B}$ -irreduciblity for any group H, where $\mathcal {B}\subseteq \mathcal {P}_f(H)$ is a right-invariant collection of finite subsets of H. When $H = F_2$ , we will be interested in the case when $\mathcal {B}$ is the collection of finite subsets of $F_2$ which are connected in the standard left Cayley graph. Section 4 gives the proof of Theorem 0.1.

1. Background

Let $\Gamma $ be a countably infinite group. Given $U, S\subseteq \Gamma $ with U finite, then we call S a (one-sided) U-spaced set if for every $g\neq h\in S$ we have $h\not \in Ug$ , and we call S a U-syndetic set if $US = \Gamma $ . A maximal U-spaced set is simply a U-spaced set which is maximal under inclusion. We remark that if S is a maximal U-spaced set, then S is $(U\cup U^{-1})$ -syndetic. We say that sets $S, T\subseteq \Gamma $ are (one-sided) U-apart if $US\cap T = \emptyset $ and $S\cap UT = \emptyset $ . Notice that much of this discussion simplifies when U is symmetric, so we will often assume this. Also, notice that the properties of being U-spaced, maximal U-spaced, U-syndetic and U-apart are all right invariant.

If A is a finite set or alphabet, then $\Gamma $ acts on $A^\Gamma $ by right shift, where given $x\in A^\Gamma $ and $g, h\in \Gamma $ , we have $(g{\cdot }x)(h) = x(hg)$ . A subshift of $A^\Gamma $ is a nonempty, closed, $\Gamma $ -invariant subset. Let $\mathrm {Sub}(A^\Gamma )$ denote the space of subshifts of $A^\Gamma $ endowed with the Vietoris topology. This topology can be described as follows. Given $X\subseteq A^\Gamma $ and a finite $U\subseteq \Gamma $ , the set of U-patterns of X is the set $P_U(X) = \{x|_U: x\in X\}\subseteq A^U$ . Then the typical basic open neighborhood of $X\in \mathrm {Sub}(A^\Gamma )$ is the set $N_U(X):= \{Y\in \mathrm {Sub}(A^\Gamma ): P_U(Y) = P_U(X)\}$ , where U ranges over finite subsets of $\Gamma $ .

A subshift $X\subseteq A^\Gamma $ is U-irreducible if for any $x_0, x_1\in X$ and any $S_0, S_1\subseteq \Gamma $ which are U-apart, there is $y\in X$ with $y|_{S_i} = x_i|_{S_i}$ for each $i< 2$ . If X is U-irreducible and $V\supseteq U$ is finite, then X is also V-irreducible. We call X strongly irreducible if there is some finite $U\subseteq \Gamma $ with $X\ U$ -irreducible. By enlarging U if needed, we can always assume U is symmetric. Let $\mathcal {S}(A^\Gamma )\subseteq \mathrm {Sub}(A^\Gamma )$ denote the set of strongly irreducible subshifts of $A^\Gamma $ , and let $\overline {\mathcal {S}}(A^\Gamma )$ denote the closure of this set in the Vietoris topology.

More generally, if $2^{\mathbb {N}}$ denotes Cantor space, then $\Gamma $ acts on $(2^{\mathbb {N}})^\Gamma $ by right shift exactly as above. If $k< \omega $ , we let $\pi _k\colon 2^{\mathbb {N}}\to 2^k$ denote the restriction to the first k entries. This induces a factor map $\tilde {\pi }_k\colon (2^{\mathbb {N}})^\Gamma \to (2^k)^\Gamma $ given by $\tilde {\pi }_k(x)(g) = \pi _k(x(g))$ ; we also obtain a map $\overline {\pi }_k\colon \mathrm {Sub}((2^{\mathbb {N}})^\Gamma )\to \mathrm {Sub}((2^k)^\Gamma )$ (where $2^k$ is viewed as a finite alphabet) given by $\overline {\pi }_k(X) = \tilde {\pi }_k[X]$ . The Vietoris topology on $\mathrm {Sub}((2^{\mathbb {N}})^\Gamma )$ is the coarsest topology making every such $\overline {\pi }_k$ continuous. We call a subflow $X\subseteq (2^{\mathbb {N}})^\Gamma $ strongly irreducible if for every $k< \omega $ , the subshift $\overline {\pi }_k(X)\subseteq (2^k)^\Gamma $ is strongly irreducible in the ordinary sense. We let $\mathcal {S}((2^{\mathbb {N}})^\Gamma )\subseteq \mathrm {Sub}((2^{\mathbb {N}})^\Gamma )$ denote the set of strongly irreducible subflows of $(2^{\mathbb {N}})^\Gamma $ , and we let $\overline {\mathcal {S}}((2^{\mathbb {N}})^\Gamma )$ denote its Vietoris closure.

The idea of considering the closure of the strongly irreducible shifts has it roots in [4]. This is made more explicit in [5], where it is shown that in $\overline {\mathcal {S}}(A^\Gamma )$ , the minimal subflows form a dense $G_\delta $ subset. More or less the same argument shows that the same holds in $\overline {\mathcal {S}}((2^{\mathbb {N}})^\Gamma )$ (see [6]). Recall that a $\Gamma $ -flow X is free if for every $g\in \Gamma \setminus \{1_\Gamma \}$ and every $x\in X$ , we have $gx\neq x$ . The main reason for considering a Cantor space alphabet is the following result, which need not be true for finite alphabets.

Proposition 1.1. In $\overline {\mathcal {S}}((2^{\mathbb {N}})^\Gamma )$ , the free flows form a dense $G_\delta $ subset.

Proof. Fixing $g\in \Gamma $ , the set $\{X\in \mathrm {Sub}((2^{\mathbb {N}})^\Gamma ): \forall x\in X\, (gx\neq x)\}$ is open; indeed, if $X_n\to X$ is a convergent sequence in $\mathrm {Sub}((2^{\mathbb {N}})^\Gamma )$ and $x_n\in X_n$ is a point fixed by g, then passing to a subsequence, we may suppose $x_n\to x\in X$ , and we have $gx = x$ . Intersecting over all $g\in \Gamma \setminus \{1_\Gamma \}$ , we see that freeness is a $G_\delta $ condition.

Thus, it remains to show that freeness is dense in $\overline {\mathcal {S}}((2^{\mathbb {N}})^\Gamma )$ . To that end, we fix $g\in \Gamma \setminus \{1_\Gamma \}$ and show that the set of shifts in $\mathcal {S}((2^{\mathbb {N}})^\Gamma )$ where g acts freely is dense. Fix $X\in \mathcal {S}((2^{\mathbb {N}})^\Gamma )$ , $k< \omega $ and a finite $U\subseteq \Gamma $ ; so a typical open set in $\mathcal {S}((2^{\mathbb {N}})^\Gamma )$ has the form $\{X'\in \mathcal {S}((2^{\mathbb {N}})^\Gamma ): P_U(\overline {\pi }_k(X')) = P_U(\overline {\pi }_k(X))\}$ . We want to produce $Y\in \mathrm {Sub}((2^{\mathbb {N}})^\Gamma )$ which is strongly irreducible, g-free and with $P_U(\overline {\pi }_k(Y)) = P_U(\overline {\pi }_k(X))$ . In fact, we will produce such a Y with $\overline {\pi }_k(Y) = \overline {\pi }_k(X)$ .

Let $D\subseteq \Gamma $ be a finite symmetric set containing g and $1_\Gamma $ . Setting $m = |D|$ , consider the subshift $\mathrm {Color}(D, m)\subseteq m^\Gamma $ defined by

$$ \begin{align*}\mathrm{Color}(D, m) := \{x\in m^\Gamma: \forall\, i < m\, [x^{-1}(\{i\})\text{ is}\ D\text{-}\text{spaced}]\}.\end{align*} $$

A greedy coloring argument shows that $\mathrm {Color}(D, m)$ is nonempty and D-irreducible. Moreover, g acts freely on $\mathrm {Color}(D, m)$ . Inject m into $2^{\{k,\ldots ,\ell -1\}}$ for some $\ell> k$ and identify $\mathrm {Color}(D, m)$ as a subflow of $(2^{\{k,\ldots ,\ell -1\}})^\Gamma $ . Then $Y:= \overline {\pi }_k(X)\times \mathrm {Color}(D, m)\subseteq (2^\ell )^\Gamma \subseteq (2^{\mathbb {N}})^\Gamma $ , where the last inclusion can be formed by adding strings of zeros to the end. Then Y is strongly irreducible, g-free and $\overline {\pi }_k(Y) = \overline {\pi }_k(X)$ .

2. UFOs and minimal subdynamics

Much of the construction will require us to reason about the product group $G\times F_2$ . So for the time being, fix countably infinite groups $\Delta \subseteq \Gamma $ . For our purposes, $\Gamma $ will be $G\times F_2$ , and $\Delta $ will be G, where we identify G with a subgroup of $G\times F_2$ in the obvious way. However, for this subsection, we will reason more generally.

Definition 2.1. Let $\Delta \subseteq \Gamma $ be countably infinite groups. A finite subset $U\subseteq \Gamma $ is called a $(\Gamma , \Delta )$ -UFO if for any maximal U-spaced set $S\subseteq \Gamma $ , we have that S meets every right coset of $\Delta $ in $\Gamma $ .

We say that the inclusion of groups $\Delta \subseteq \Gamma $ admits UFOs if for every finite $U\subseteq \Gamma $ , there is a finite $V\subseteq \Gamma $ with $V\supseteq U$ which is a $(\Gamma , \Delta )$ -UFO.

As a word of caution, we note that the property of being a $(\Gamma , \Delta )$ -UFO is not upwards closed.

The terminology comes from considering the case of a product group, that is, $\Gamma = \mathbb {Z}\times \mathbb {Z}$ and $\Delta = \mathbb {Z}\times \{0\}$ . Figure 1 depicts a typical UFO subset of $\mathbb {Z}\times \mathbb {Z}$ .

Figure 1 Sighting in Roswell; a $(\mathbb {Z}\times \mathbb {Z}, \mathbb {Z}\times \{0\})$ -UFO subset of $\mathbb {Z}\times \mathbb {Z}$ .

Proposition 2.2. Let $\Delta $ be a subgroup of $\Gamma $ . If $|\bigcap _{u \in U} u \Delta u^{-1}|$ is infinite for every finite set $U \subseteq \Gamma $ , then $\Delta \subseteq \Gamma $ admits UFOs. In particular, if $\Delta $ contains an infinite subgroup that is normal in $\Gamma $ , then $\Delta \subseteq \Gamma $ admits UFOs.

Proof. We prove the contrapositive. So assume that $\Delta \subseteq \Gamma $ does not admit UFOs. Let $U \subseteq \Gamma $ be a finite symmetric set such that no finite $V \subseteq \Gamma $ containing U is a $(\Gamma , \Delta )$ -UFO. Let $D \subseteq \Delta $ be finite, symmetric and contain the identity. It will suffice to show that $C = \bigcap _{u \in U} u D u^{-1}$ satisfies $|C| \leq |U|$ .

Set $V = U \cup D^2$ . Since V is not a $(\Gamma , \Delta )$ -UFO, there is a maximal V-spaced set $S \subseteq \Gamma $ and $g \in \Gamma $ with $S \cap \Delta g = \varnothing $ . Since S is V-spaced and $u^{-1} C^2 u \subseteq D^2 \subseteq V$ , the set $C_u = (u S) \cap (C g)$ is $C^2$ -spaced for every $u \in U$ . Of course, any $C^2$ -spaced subset of $C g$ is empty or a singleton, so $|C_u| \leq 1$ for each $u \in U$ . On the other hand, since S is maximal we have $V S = \Gamma $ , and since $S \cap \Delta g = \varnothing $ we must have $C g \subseteq U S$ . Therefore, $|C| = |C g| = \sum _{u \in U} |C_u| \leq |U|$ .

In the spaces $\overline {\mathcal {S}}(k^\Gamma )$ and $\overline {\mathcal {S}}((2^{\mathbb {N}})^\Gamma )$ , the minimal flows form a dense $G_\delta $ . However, when $\Delta \subseteq \Gamma $ is a subgroup, we can ask about the properties of members of $\overline {\mathcal {S}}(k^\Gamma )$ and $\overline {\mathcal {S}}((2^{\mathbb {N}})^\Gamma )$ viewed as $\Delta $ -flows.

Definition 2.3. Given a subshift $X\subseteq k^\Gamma $ and a finite $E\subseteq \Gamma $ , we say that X is $(\Delta , E)$ -minimal if for every $x\in X$ and every $p\in P_E(X)$ , there is $g\in \Delta $ with $(gx)|_E = p$ . Given a subflow $X\subseteq (2^{\mathbb {N}})^\Gamma $ and $n\in \mathbb {N}$ , we say that X is $(\Delta , E, n)$ -minimal if $\overline {\pi }_n(X)\subseteq (2^n)^\Gamma $ is $(\Delta , E)$ -minimal. When $\Delta = \Gamma $ , we simply say that X is E-minimal or $(E, n)$ -minimal.

The set of $(\Delta , E)$ -minimal flows is open in $\mathrm {Sub}(k^\Gamma )$ , and $X\subseteq k^\Gamma $ is minimal as a $\Delta $ -flow iff it is $(\Delta , E)$ -minimal for every finite $E\subseteq \Gamma $ . Similarly, the set of $(\Delta , E, n)$ -minimal flows is open in $\mathrm {Sub}((2^{\mathbb {N}})^\Gamma )$ , and $X\subseteq (2^{\mathbb {N}})^\Gamma $ is minimal as a $\Delta $ -flow iff it is $(\Delta , E, n)$ minimal for every finite $E\subseteq \Gamma $ and every $n\in \mathbb {N}$ .

In the proof of Proposition 2.4, it will be helpful to extend conventions about the shift action to subsets of $\Gamma $ . If $U\subseteq \Gamma $ , $g\in G$ and $p\in k^U$ , we write $g{\cdot }p\in k^{Ug^{-1}}$ for the function where given $h\in Ug^{-1}$ , we have $(g{\cdot }p)(h) = p(hg)$ .

Proposition 2.4. Suppose $\Delta \subseteq \Gamma $ are countably infinite groups and that $\Delta \subseteq \Gamma $ admits UFOs. Then $\{X\in \overline {\mathcal {S}}(k^\Gamma ): X \text { is minimal as a}\ \Delta -\text {flow}\}$ is a dense $G_\delta $ subset. Similarly, $\{X\in \overline {\mathcal {S}}(2^{\mathbb {N}})^\Gamma : X\text { is minimal as a}\ \Delta -\text {flow}\}$ is a dense $G_\delta $ subset.

Proof. We give the arguments for $k^\Gamma $ , as those for $(2^{\mathbb {N}})^\Gamma $ are very similar.

It suffices to show for a given finite $E\subseteq \Gamma $ that the collection of $(\Delta , E)$ -minimal flows is dense in $\overline {\mathcal {S}}(k^\Gamma )$ . By enlarging E if needed, we can assume that E is symmetric.

Consider a nonempty open $O\subseteq \overline {\mathcal {S}}(k^\Gamma )$ . By shrinking O and/or enlarging E if needed, we can assume that for some $X\in \mathcal {S}(k^\Gamma )$ , we have $O = N_E(X)\cap \overline {\mathcal {S}}(k^\Gamma )$ . We will build a $(\Delta , E)$ -minimal shift Y with $Y\in N_E(X)\cap \mathcal {S}(k^\Gamma )$ . Fix a finite symmetric $D\subseteq \Gamma $ so that X is D-irreducible. Then fix a finite  $U\subseteq \Gamma $ which is large enough to contain an $EDE$ -spaced set $Q\subseteq U\cap \Delta $ of cardinality $|P_E(X)|$ , and enlarging U if needed, choose such a Q with $EQ\subseteq U$ . Fix a bijection $Q\to P_E(X)$ by writing $P_E(X) = \{p_g: g\in Q\}$ . Because X is D-irreducible, we can find $\alpha \in P_U(X)$ so that $(gq)|_E = p_g$ for every $g\in Q$ . By Proposition 2.2, fix a finite $V\subseteq \Gamma $ with $V\supseteq UDU$ which is a $(\Gamma , \Delta )$ -UFO. We now form the shift

$$ \begin{align*}Y = \{y\in X: \exists\text{ a max.}\ V\text{-}\text{spaced set}\ T\text{ so that }\forall g\in T\, (g{\cdot}y)|_U = \alpha\}.\end{align*} $$

Because $V = UDU$ and X is D-irreducible, we have that $Y\neq \emptyset $ . In particular, for any maximal V-spaced set $T\subseteq \Gamma $ , we can find $y\in Y$ so that $(gy)|_U = \alpha $ for every $g\in T$ . We also note that $Y\in N_E(X)$ by our construction of $\alpha $ .

To see that Y is $(\Delta , E)$ -minimal, fix $y\in Y$ and $p\in P_E(Y)$ . Suppose this is witnessed by the maximal V-spaced set $T\subseteq \Gamma $ . Because V is a $(\Gamma , \Delta )$ -UFO, find $h\in \Delta \cap T$ . So $(hy)|_U = \alpha $ . Now, suppose $g\in Q$ is such that $p = p_g$ . We have $(ghy)|_E = (g\cdot ((hy)|_U)|_E = p_g$ .

To see that $Y\in \mathcal {S}(k^\Gamma )$ , we will show that Y is $DUVUD$ -irreducible. Suppose $y_0, y_1\in Y$ and $S_0, S_1\subseteq \Gamma $ are $DUVUD$ -apart. For each $i< 2$ , fix $T_i\subseteq \Gamma $ a maximal V-spaced set which witnesses that $y_i$ is in Y. Set $B_i = \{g\in T_i: DUg\cap S_i\neq \emptyset \}$ . Notice that $B_i\subseteq UDS_i$ . It follows that $B_0\cup B_1$ is V-spaced, so extend to a maximal V-spaced set B. It also follows that $S_i\cup UB_i\subseteq U^2DS_i$ . Since $V\supseteq UDU$ and by the definition of $B_i$ , the collection of sets $\{S_i\cup UB_i: i< 2\}\cup \{Ug: g\in B\setminus (B_0\cup B_1)\}$ is pairwise D-apart. By the D-irreducibility of X, we can find $y\in X$ with $y|_{S_i\cup UB_i} = y_i|_{S_i\cup UB_i}$ for each $i< 2$ and with $(gy)|_U = \alpha $ for each $g\in B\setminus (B_0\cup B_1)$ . Since $B_i\subseteq T_i$ , we actually have $(gy)|_U = \alpha $ for each $g\in B$ . So $y\in Y$ and $y|_{S_i} = y_i|_{S_i}$ as desired.

Proof of Theorem 0.3.

By Proposition 2.4, the generic member of $\overline {\mathcal {S}}((2^{\mathbb {N}})^\Gamma )$ is minimal as a $\Delta _n$ -flow for each $n\in \mathbb {N}$ , and by Proposition 1.1, the generic member of $\overline {\mathcal {S}}((2^{\mathbb {N}})^\Gamma $ is free.

In contrast to Theorem 0.1, the next example shows that Question 0.2 is nontrivial to answer in full generality.

Theorem 2.5. Let $G=\sum _{\mathbb {N}} (\mathbb {Z}/2\mathbb {Z})$ , and let X be a G flow with infinite underlying space. Then there exists an infinite subgroup H such that X is not minimal as an H flow.

Proof. We may assume that X is a minimal G-flow, as otherwise we may take $H = G$ . We construct a sequence $X\supsetneq K_0\supseteq K_1\supseteq \cdots $ of proper, nonempty, closed subsets of X and a sequence of group elements $\{g_n: n\in \mathbb {N}\}$ so that by setting $K = \bigcap _{\mathbb {N}} K_n$ and $H = \langle g_n: n\in \mathbb {N}\rangle $ , then K will be a minimal H-flow. Start by fixing a closed, proper subset $K_0\subsetneq X$ with nonempty interior. Suppose $K_n$ has been created and is $\langle g_0,\ldots ,g_{n-1}\rangle $ -invariant. As X is a minimal G-flow, the set $S_n:= \{g\in G: \mathrm {Int}(gK_n\cap K_n)\neq \emptyset \}$ is infinite. Pick any $g_n\in S_n\setminus \{1_G\}$ , and set $K_{n+1} = g_nK_n\cap K_n$ . As $g_n^2 = 1_G$ , we see that $K_{n+1}$ is $g_n$ -invariant, and as G is abelian, we see that $K_{n+1}$ is also $g_i$ -invariant for each $i< n$ . It follows that K will be H-invariant as desired.

Before moving on, we give a conditional proof of Theorem 0.1, which works as long as some nonamenable group admits a strongly irreducible shift without an invariant measure. It is the inspiration for our overall construction.

Proposition 2.6. Let G and H be countably infinite groups, and suppose that for some $k< \omega $ and some strongly irreducible flow $Y\subseteq k^H$ that Y does not admit an H-invariant measure. Then there is a minimal G-flow which does not admit a characteristic measure.

Proof. Viewing $Z = k^G\times Y$ as a subshift of $k^{G\times H}$ , then Z is strongly irreducible and does not admit an H-invariant probability measure. The property of not possessing an H-invariant measure is an open condition in $\mathrm {Sub}(k^{G\times H})$ ; indeed, if $X_n\to X$ is a convergent sequence in $\mathrm {Sub}(k^{G\times H})$ and $\mu _n$ is an H-invariant probability measure supported on $X_n$ , then by passing to a subsequence, we may suppose that the $\mu _n$ weak $^*$ -converge to some H-invariant probability measure $\mu $ supported on X. By Proposition 2.4, we can therefore find $X\subseteq k^{G\times H}$ which is minimal as a G-flow and which does not admit an H-invariant measure. As H acts by G-flow automorphisms on X, we see that X does not admit a characteristic measure.

Unfortunately, the question of if there exists any countable group H and a strongly irreducible H-subshift Y with no H-invariant measure is an open problem. Therefore, our construction proceeds by considering the free group $F_2$ and defining a suitable weakening of strongly irreducible subshift which is strong enough for G-minimality to be generic in $(G\times F_2)$ -subshifts but weak enough for $(G\times F_2)$ -subshifts without $F_2$ -invariant measures to exist.

3. Variants of strong irreducibility

In this section, we investigate a weakening of strong irreducibility that one can define given any right-invariant collection $\mathcal {B}$ of finite subsets of a given countable group. For our overall construction, we will consider $F_2$ and $G\times F_2$ , but we give the definitions for any countably infinite group $\Gamma $ . Write $\mathcal {P}_f(\Gamma )$ for the collection of finite subsets of $\Gamma $ .

Definition 3.1. Fix a right-invariant subset $\mathcal {B}\subseteq \mathcal {P}_f(\Gamma )$ . Given $k\in \mathbb {N}$ , we say that a subshift $X\subseteq k^\Gamma $ is $\mathcal {B}$ -irreducible if there is a finite $D\subseteq \Gamma $ so that for any $m< \omega $ , any $B_0,\ldots , B_{m-1}\in \mathcal {B}$ , and any $x_0,\ldots ,x_{m-1}\in X$ , if the sets $\{B_0,\ldots , B_{m-1}\}$ are pairwise D-apart, then there is $y\in X$ with $y|_{B_i} = x_i|_{B_i}$ for each $i< m$ . We call D the witness to $\mathcal {B}$ -irreducibility. If we have D in mind, we can say that X is $\mathcal {B}$ -D-irreducible.

We say that a subflow $X\subseteq (2^{\mathbb {N}})^\Gamma $ is $\mathcal {B}$ -irreducible if for each $k\in \mathbb {N}$ , the subshift $\overline {\pi }_k(X)\subseteq (2^k)^\Gamma $ is $\mathcal {B}$ -irreducible.

We write $\mathcal {S}_{\mathcal {B}}(k^\Gamma )$ or $\mathcal {S}_{\mathcal {B}}((2^{\mathbb {N}})^\Gamma )$ for the set of $\mathcal {B}$ -irreducible subflows of $k^\Gamma $ or $(2^{\mathbb {N}})^\Gamma $ , respectively, and we write $\overline {\mathcal {S}}_{\mathcal {B}}(k^\Gamma )$ or $\overline {\mathcal {S}}_{\mathcal {B}}((2^{\mathbb {N}})^\Gamma )$ for the Vietoris closures.

Remark.

1. 1. If $\mathcal {B}$ is closed under unions, it is enough to consider $m = 2$ . However, this will often not be the case.

2. 2. By compactness, if $X\subseteq k^\Gamma $ is $\mathcal {B}$ -D-irreducible, $\{B_n: n< \omega \}\subseteq \mathcal {B}$ is pairwise D-apart, and $\{x_n: n< \omega \}\subseteq X$ , then there is $y\in X$ with $y|_{B_i} = x_i|_{B_i}$ .

3. 3. If $\mathcal {B}\subseteq \mathcal {B}'$ , then $\mathcal {S}_{\mathcal {B}'}(k^\Gamma )\subseteq \mathcal {S}_{\mathcal {B}}(k^\Gamma )$ and $\mathcal {S}_{\mathcal {B}'}((2^{\mathbb {N}})^\Gamma )\subseteq \mathcal {S}_{\mathcal {B}}((2^{\mathbb {N}})^\Gamma )$

When $\mathcal {B}$ is the collection of all finite subsets of H, then we recover the notion of a strongly irreducible shift. Again, we consider Cantor space alphabets to obtain freeness.

Proposition 3.2. For any right-invariant collection $\mathcal {B}\subseteq \mathcal {P}_f(\Gamma )$ , the generic member of $\overline {\mathcal {S}}_{\mathcal {B}}((2^{\mathbb {N}})^\Gamma )$ is free.

Proof. Analyzing the proof of Proposition 1.1, we see that the only properties that we need of the collections $\mathcal {S}_{\mathcal {B}}(k^\Gamma )$ and $\mathcal {S}_{\mathcal {B}}((2^{\mathbb {N}})^\Gamma )$ for the proof to generalize are that they are closed under products and contain the flows $\mathrm {Color}(D, m)$ . If $k, \ell \in \mathbb {N}$ an $X\subseteq k^\Gamma $ and $Y\subseteq \ell ^\Gamma $ are $\mathcal {B}$ -D-irreducible and $\mathcal {B}$ -E-irreducible for some finite $D, E\subseteq \Gamma $ , then $X\times Y\subseteq (k\times \ell )^\Gamma $ will be $\mathcal {B}$ - $(D\cup E)$ -irreducible. And as $\mathrm {Color}(D, m)$ is strongly irreducible, it is $\mathcal {B}$ -irreducible.

Now, we consider the group $F_2$ . We consider the left Cayley graph of $F_2$ with respect to the standard generating set $A:= \{a, b, a^{-1}, b^{-1}\}$ . We let $d\colon F_2\times F_2\to \omega $ denote the graph metric. Write $D_n = \{s\in F_2: d(s, 1_{F_2}) \leq n\}$ .

Definition 3.3. Given n with $1\leq n< \omega $ , we set

$$ \begin{align*}\mathcal{B}_n = \{D\in \mathcal{P}_f(F_2): \text{ connected components of}\ D\ \text{are pairwise}\ D_n\text{-}\text{apart}\}.\end{align*} $$

Write $\mathcal {B}_\omega $ for the collection of finite, connected subsets of $F_2$ .

Proposition 3.4. Suppose $X\subseteq k^{F_2}$ is $\mathcal {B}_\omega $ -irreducible. Then there is some $n< \omega $ for which X is $\mathcal {B}_n$ -irreducible.

Proof. Suppose X is $\mathcal {B}_\omega $ - $D_n$ -irreducible. We claim X is $\mathcal {B}_n$ - $D_n$ -irreducible. Suppose $m< \omega $ , $B_0,\ldots ,B_{m-1}\in \mathcal {B}_n$ are pairwise $D_n$ -apart, and $x_0,\ldots ,x_{m-1}\in X$ . For each $i< m$ , we suppose $B_i$ has $n_i$ -many connected componenets, and we write $\{C_{i,j}: j< n_i\}$ for these components. Then the collection of connected sets $\bigcup _{i< m} \{C_{i,j}: j< n_i\}$ is pairwise $D_n$ -apart. As X is $\mathcal {B}_\omega $ - $D_n$ -irreducible, we can find $y\in X$ so that for each $i< m$ and $j< n_i$ , we have $y|_{C_{i,j}} = x_i|_{C_{i,j}}$ . Hence, $y|_{B_i} = x_i|_{B_i}$ , showing that X is $\mathcal {B}_n$ - $D_n$ -irreducible.

We now construct a $\mathcal {B}_\omega $ -irreducible subshift with no $F_2$ -invariant measure. We consider the alphabet $A^2$ and write $\pi _0, \pi _1\colon A^2\to A$ for the projections. We set

$$ \begin{align*} X_{pdox} = \{x\in (A^2)^{F_2}:\, &\forall g, h\in F_2 \, \forall i, j< 2 \\ &(i, g)\neq (j, h)\Rightarrow \pi_i(x(g))\cdot g\neq \pi_j(x(h))\cdot h\}. \end{align*} $$

More informally, the flow $X_{pdox}$ is the space of ‘ $2$ -to- $1$ paradoxical decompositions’ of $F_2$ using A. We remark that here, our decomposition need not be a partition of $F_2$ ; we just ask for disjoint $S_0, S_1\subseteq F_2$ such that for every $g\in G$ and $i< 2$ , we have $Ag\cap S_i\neq \emptyset $ . This is in some sense the prototypical example of an $F_2$ -shift with no $F_2$ -invariant measure.

Lemma 3.5. $X_{pdox}$ has no $F_2$ -invariant measure.

Proof. For $u \in A^2$ set $Y_u = \{x \in X_{pdox} : x(1_G) = u\}$ . Notice that if $y \in Y_u$ , $i < 2$ and $x = \pi _i(u) y$ , then $x(\pi _i(u)^{-1}) = y(1_G) = u$ . Consequently, if $u, v\in A^2$ , $x \in \pi _i(u) Y_u \cap \pi _j(v) Y_v$ then, since $x \in X_{pdox}$ and

$$ \begin{align*}\pi_i(x(\pi_i(u)^{-1})) \pi_i(u)^{-1} = 1_G = \pi_j(x(\pi_j(v)^{-1})) \pi_j(v)^{-1},\end{align*} $$

we must have that $(i, \pi _i(u)) = (j, \pi _j(v))$ , and hence also

$$ \begin{align*}\pi_{1-i}(u) = \pi_{1-i}(x(\pi_i(u)^{-1})) = \pi_{1-j}(x(\pi_j(v)^{-1})) = \pi_{1-j}(v).\end{align*} $$

Therefore, $\pi _i(u) Y_u \cap \pi _j(v) Y_v = \varnothing $ whenever $(i,u) \neq (j,v)$ .

If $\mu $ were an invariant Borel probability measure on $X_{pdox}$ , then we would have

$$ \begin{align*}2 \mu(X_{pdox}) = 2 \sum_{u \in A^2} \mu(Y_u) = \sum_{i<2} \sum_{u \in A^2} \mu(\pi_i(u) Y_u) \leq \mu(X)\end{align*} $$

which is a contradiction.

When proving that $X_{pdox}$ is $\mathcal {B}_\omega $ -irreducible, note that $D_1 = A\cup \{1_{F_2}\}$ .

Proposition 3.6. $X_{pdox}$ is $\mathcal {B}_\omega $ - $D_4$ -irreducible.

Proof. The proof will use a $2$ -to- $1$ instance of Hall’s matching criterion [7] which we briefly describe. Fix a bipartite graph $\mathbb {G} = (V, E)$ with partition $V = V_0\sqcup V_1$ . Given $S\subseteq V_0$ , write $N_{\mathbb {G}}(S) = \{v\in V_1: \exists u\in S (u, v)\in E\}$ . Then the matching condition we need states that if for every finite $S\subseteq V_0$ , we have $|N_{\mathbb {G}}(S)|\geq 2S$ , then there is $E'\subseteq E$ so that in the graph $\mathbb {G}':= (V, E')$ , $d_{\mathbb {G}'}(u) = 2$ for every $u\in V_0$ .

Let $B_0,\ldots ,B_{k-1}\in \mathcal {B}_\omega $ be pairwise $D_4$ -apart. Let $x_0,\ldots ,x_{k-1}\in X_{pdox}$ . To construct $y\in X_{pdox}$ with $y|_{B_i} = x_i|_{B_i}$ for each $i< k$ , we need to verify a $2$ -to- $1$ Hall’s matching criterion on every finite subset of $F_2\setminus \bigcup _{i< k} B_i$ . Call $s\in F_2$ matched if for some $i< k$ , some $g\in B_i$ and some $j< 2$ , we have $s = \pi _j(x_i(g))\cdot g$ . So we need for every finite $E\in \mathcal {P}_f(F_2\setminus \bigcup _{i<k} B_i)$ that $AE$ contains at least $2|E|$ -many unmatched elements. Towards a contradiction, let $E\in \mathcal {P}_f(F_2\setminus \bigcup _{i<k} B_i)$ be a minimal failure of the Hall condition.

In the left Cayley graph of $F_2$ , given a reduced word w in alphabet $A = \{a, b, a^{-1}, b^{-1}\}$ , write $N_w$ for the set of reduced words which end with w. Now, find $t\in E$ (let us assume the leftmost character of t is a) so that all of $E\cap N_{at}$ , $E\cap N_{bt}$ and $E\cap N_{b^{-1}t}$ are empty. If any two of $at$ , $bt$ and $b^{-1}t$ is an unmatched point in $AE$ , then $E\setminus \{t\}$ is a smaller failure of Hall’s criterion. So there must be some $i< k$ , some $g\in B_i$ and some $j< 2$ , we have $\pi _j(x_i(g))\cdot g \in \{at, bt, b^{-1}t\}$ . Let us suppose $\pi _j(x_i(g))\cdot g = at$ . Note that since $g\not \in E$ , we must have $g\in \{bat, a^2t, b^{-1}at\}$ . But then since $B_i$ is connected, we have $D_1B_i\cap \{bt, b^{-1}t\} = \emptyset $ , and since the other $B_q$ are at least distance $5$ from $B_i$ , we have $D_1B_q\cap \{bt, b^{-1}t\} = \emptyset $ for every $q\in k\setminus \{i\}$ . In particular, $bt$ and $b^{-1}t$ are unmatched points in $AE$ , a contradiction.

We remark that $X_{pdox}$ is not $D_n$ -irreducible for any $n\in \mathbb {N}$ . See Figure 2.

Figure 2 A pair of outgoing edges, drawn in solid red, is chosen at each of $v_{00}$ , $v_{01}$ , $v_{10}$ and $v_{11}$ . Edges which must consequently be oriented in a particular direction are indicated with dashed red arrows. Most importantly, $v_{\varnothing }$ is forced to direct an edge to $u_\varnothing $ . By considering the generalization of this picture for any length of binary string, we see that $X_{pdox}$ cannot be $D_n$ -irreducible for any $n\in \mathbb {N}$ .

4. The construction

Our goal for the rest of the paper is to use $X_{pdox}$ to build a subflow of $(2^{\mathbb {N}})^{G\times F_2}$ which is free, G-minimal and with no $F_2$ -invariant measure. In what follows, given an $F_2$ -coset $\{g\}\times F_2$ , we endow this coset with the left Cayley graph for $F_2$ using the generating set A exactly as above. We extend the definition of $\mathcal {B}_n$ to refer to finite subsets of any given $F_2$ -coset.

Definition 4.1. Given n with $1\leq n\leq \omega $ , we set

$$ \begin{align*}\mathcal{B}_n^*= \{D\in \mathcal{P}_f(G\times F_2): \text{ for each}\ F_2-\text{coset}\ C, D\cap C\in \mathcal{B}_n\}.\end{align*} $$

Given $y\in k^{G\times F_2}$ and $g\in G$ , we define $y_g\in k^{F_2}$ where given $s\in F_2$ , we set $y_g(s) = y(g, s)$ . If $X\subseteq k^{F_2}$ is $\mathcal {B}_n$ -irreducible, then the subshift $X^G\subseteq k^{G\times F_2}$ is in $\mathcal {S}_{\mathcal {B}_n^*}$ , where we view $X^G$ as the set $\{y\in k^{G\times F_2}: \forall g\in G\, (y_g\in X)\}$ . In particular, $(X_{pdox})^G$ is $\mathcal {B}^*_4$ -irreducible. By encoding $(X_{pdox})^G$ as a subshift of $(2^m)^{G\times F_2}$ for some $m\in \mathbb {N}$ and considering $\tilde {\pi }_m^{-1}((X_{pdox})^G)\subseteq (2^{\mathbb {N}})^{G\times F_2}$ , we see that there is a $\mathcal {B}_4^*$ -irreducible subflow of $(2^{\mathbb {N}})^{G\times F_2}$ for which the $F_2$ -action doesn’t fix a measure. It follows that such subflows constitute a nonempty open subset of $\Phi := \overline {\bigcup _n \mathcal {S}_{\mathcal {B}_n^*}((2^{\mathbb {N}})^{G\times F_2})}$ . Combining the next result with Proposition 3.2, we will complete the proof of Theorem 0.1.

Proposition 4.2. With $\Phi $ as above, the G-minimal flows are dense $G_\delta $ in $\Phi $ .

Proof. We show the result for $\Phi _k:= \overline {\bigcup _n \mathcal {S}_{\mathcal {B}_n^*}(k^{G\times F_2})}$ to simplify notation; the proof in full generality is almost identical.

We only need to show density. To that end, fix a finite symmetric $E\subseteq G\times F_2$ which is connected in each $F_2$ -coset. It is enough to show that the $(G, E)$ -minimal subshifts are dense in $\Phi _k$ . Fix some nonempty open $O\subseteq \Phi _k$ . By enlarging E and/or shrinking O, we may assume that for some $n< \omega $ and $X\in \mathcal {S}_{\mathcal {B}_n^*}(k^{G\times F_2})$ that $O = \{X'\in \Phi _k: P_E(X') = P_E(X)\}$ . We will build a $(G, E)$ -minimal subshift $Y\subseteq k^{G\times F_2}$ so that $P_E(Y) = P_E(X)$ and so that for some $N< \omega $ , we have $Y\in \mathcal {S}_{\mathcal {B}_N^*}(k^{G\times F_2})$ .

Recall that $D_n\subseteq F_2$ denotes the ball of radius n. Fix a finite, symmetric $D\subseteq G\times F_2$ so that $\{1_G\}\times D_{2n}\subseteq D$ and X is $\mathcal {B}_n^*$ -D-irreducible. Find a finite symmetric $U_0\subseteq G$ with $1_G\subseteq U_0$ and $r< \omega $ so that upon setting $U = U_0\times D_r\subseteq G\times F_2$ , then U is large enough to contain an $EDE$ -spaced set $Q\subseteq G$ with $EQ\subseteq U$ . As X is $\mathcal {B}_n^*$ -D-irreducible, there is a pattern $\alpha \in P_U(X)$ so that $\{(g\alpha )|_E: g\in Q\} = P_E(X)$ .

Let $V\supseteq UD^2U$ be a $(G\times F_2, G)$ -UFO. We remark that for most of the remainder of the proof, it would be enough to have $V\supseteq UDU$ ; we only use the stronger assumption $V\supseteq UD^2U$ in the proof of the final claim. Consider the following subshift:

$$ \begin{align*}Y = \{y\in X: \exists\text{ a max.}\ V\text{-spaced set}\ T\ \text{so that }\forall g\in T\, (gy)|_U = \alpha\}.\end{align*} $$

The proof that Y is nonempty and $(G, E)$ -minimal is exactly the same as the analogous proof from Proposition 2.4. Note that by construction, we have $P_E(Y) = P_E(X)$ .

We now show that $Y\in \mathcal {S}_{\mathcal {B}_N^*}(k^{G\times F_2})$ for $N = 4r+3n$ . Set $W = DUVUD$ . We show that Y is $\mathcal {B}_N^*$ -W-irreducible. Suppose $m< \omega $ , $y_0,\ldots ,y_{m-1}\in Y$ and $S_0,\ldots ,S_{m-1}\in \mathcal {B}_N^*$ are pairwise W-apart. Suppose for each $i< m$ that $T_i\subseteq G\times F_2$ is a maximal V-spaced set which witness that $y_i\in Y$ . Set $B_i = \{g\in T_i: DUg\cap S_i\neq \emptyset \}$ . Then $\bigcup _{i< m} B_i$ is V-spaced, so enlarge to a maximal V-spaced set $B\subseteq G\times F_2$ .

For each $i< m$ , we enlarge $S_i\cup UB_i$ to $J_i\in \mathcal {B}_n^*$ as follows. Suppose $C\subseteq G\times F_2$ is an $F_2$ -coset. Each set of the form $C\cap Ug$ is connected. Since $S_i\in \mathcal {B}_N^*$ , it follows that given $g\in B_i$ , there is at most one connected component $\Theta _{C, g}$ of $S_i\cap C$ with $Ug\cap \Theta _{C, g} = \emptyset $ , but $Ug\cap D_n\Theta _{C, g}\neq \emptyset $ . We add the line segment in C connecting $\Theta _{C, g}$ and $Ug$ . Upon doing this for each $g\in B_i$ and each $F_2$ -coset C, this completes the construction of $J_i$ . Observe that $J_i\subseteq D_{n-1}S_i\cap UB_i$ .

Claim. Let C be an $F_2$ -coset, and suppose $Y_0$ is a connected component of $S_i\cap C$ . Let Y be the connected component of $J_i\cap C$ with $Y_0\subseteq Y$ . Then $Y\subseteq D_{2r+n}Y_0$ . In particular, if $Y_0\neq Z_0$ are two connected components of $S_i\cap C$ , then $Y_0$ and $Z_0$ do not belong to the same component of $J_i\cap C$ .

Proof. Let $L = \{x_j: j< \omega \}\subseteq C$ be a ray with $x_0\in Y_0$ and $x_j\not \in Y_0$ for any $j\geq 1$ . Then $\{j< \omega : x_j\in J_i\cap C\}$ is some finite initial segment of $\omega $ . We want to argue that for some $j\leq 2r+n+1$ , we have $x_j\not \in J_i\cap C$ . First, we argue that if $x_n\in J_i\cap C$ , then $x_n\in UB_i$ . Otherwise, we must have $x_n\in D_{n-1}S_i$ . But since $x_n\not \in D_{n-1}Y_0$ , there must be another component $Y_1$ of $S_i\cap C$ with $x_n\in D_nY_1$ . But this implies that $Y_0$ and $Y_1$ are not $D_{2n-1}$ -apart, a contradiction since $2n-1\leq 4r-3n = N$ .

Fix $g\in B_i$ with $x_n\in Ug$ . Let $q< \omega $ be least with $q> n$ and $x_q \not \in U_g$ . We must have $q\leq 2r+n+1$ . We claim that $x_q\not \in J_i\cap C$ . Towards a contradiction, suppose $x_q\in J_i\cap C$ . We cannot have $x_q\in UB_i$ , so we must have $x_q\in D_{n-1}S_i$ . But now there must be some component $Y_1$ of $S_i\cap C$ with $x_q\in D_{n-1}Y_1$ . But then $D_{2r+2n}Y_0\cap Y_1\neq \emptyset $ , a contradiction as $Y_0$ and $Y_1$ are $D_N$ -apart. This concludes the proof that $Y\subseteq D_{2r+n}Y_0$ .

Now, suppose $Y_0\neq Z_0$ are two connected components of $S_i\cap C$ . Then $Y_0$ and $Z_0$ are N-apart. In particular, $Z_0\not \subseteq D_{2r+n}Y_0$ , so cannot belong to the same connected component of $J_i\cap C$ as $Y_0$ .

Claim. $J_i\in \mathcal {B}_n^*$ .

Proof. Fix an $F_2$ -coset C and two connected components $Y\neq Z$ of $J_i\cap C$ . By the previous claim, each of Y and Z can only contain at most one nonempty component of $S_i\cap C$ . The claim will be proven after considering three cases.

1. 1. First, suppose each of Y and Z contain a nonempty component of $S_i\cap C$ , say $Y_0\subseteq Y$ and $Z_0\subseteq Z$ . Then since $Y_0$ and $Z_0$ are $D_{4r+3n}$ -apart, the previous claim implies that Y and Z are $D_n$ -apart.

2. 2. Now, suppose Y contains a nonempty component $Y_0$ of $S_i\cap C$ and that Z does not. Then for some $g\in B_i$ , we have $Z = Ug\cap C$ . Towards a contradiction, suppose $D_nY\cap Ug \neq \emptyset $ . Let $L = \{x_j: j\leq M\}$ be the line segment connecting Y and $Ug$ with $L\cap Y = \{x_0\}$ and $L\cap Ug = \{x_M\}$ . We must have $M\leq n$ . We cannot have $x_0\in UB_i$ , so we must have $x_0\in D_{n-1}S_i$ . This implies that $x_0\in D_{n-1}Y_0$ . We cannot have $x_0\in Y_0$ , as otherwise, we would have connected $Y_0$ and $Ug\cap C$ when constructing $J_i$ . It follows that for some $h\in B_i$ , we have that $x_0$ is on the line segment $L' = \{x_j': j\leq M'\}$ connecting $Y_0$ and $Uh\cap C$ , and we have $M'\leq n$ . But this implies that $Ug\cap D_{2n}Uh\neq \emptyset $ , a contradiction since $V\supseteq UDU$ and $D\supseteq D_{2n}$ .

3. 3. If neither Y nor Z contain a component of $S_i\cap C$ , then there are $g\neq h\in B_i$ with $Y = Uh\cap C$ and $Z = Ug\cap C$ . It follows that Y and Z are $D_n$ -apart.

Claim. Suppose $i\neq j< m$ . Then $J_i$ and $J_j$ are D-apart.

Proof. We have that $J_i\subseteq D_{n-1}S_i\cup UB_i$ , and likewise for j. As $UB_i\subseteq U^2DS_i$ and as $D\supseteq D_{2n}$ , we have $J_i\subseteq U^2DS_i$ , and likewise for j. As $S_i$ and $S_j$ are W-apart and as $V\supseteq UDU$ , we see that $J_i$ and $J_j$ are D-apart.

Claim. Suppose $g\in B\setminus \bigcup _{i< m} B_i$ . Then $Ug$ and $J_i$ are D-apart for any $i< m$ .

Proof. As $g\not \in B_i$ , we have $U_g$ and $S_i$ are D-apart. Also, for any $h\in B$ with $g\neq h$ , we have that $Ug$ and $Uh$ are D-apart. Now, suppose $DUg\cap J_i\neq \emptyset $ . If $x\in DUg\cap J_i$ , then on the coset $C = F_2x$ , x must belong on the line between a component of $S_i\cap C$ and $Uh$ for some $h\in B_i$ . Furthermore, we have $x\in D_{n-1}Uh$ . But since $D_{2n}\subseteq D$ , this contradicts that $Ug$ and $Uh$ are $D^2$ -apart (using the full assumption $V\supseteq UD^2U$ ).

We can now finish the proof of Proposition 4.2. The collection $\{J_i: i< m\}\cup \{Ug: g\in B\setminus (\bigcup _{i< m} B_i)\}$ is a pairwise D-apart collection of members of $\mathcal {B}_n^*$ . As X is $\mathcal {B}_n^*$ -D-irreducible, we can find $y\in X$ with $y|_{J_i} = y_i|_{J_i}$ for each $i< m$ and with $(gy)|_U = \alpha $ for each $g\in B\setminus (\bigcup _{i< m} B_i)$ . As $J_i\supseteq UB_i$ and since $B_i\subseteq T_i$ , we actually have $(gy)|_U = \alpha $ for each $g\in B$ . As B is a maximal V-spaced set, it follows that $y\in Y$ and $y|_{S_i} = y_i|_{S_i}$ as desired.