Mean Topological Dimension for Actions of Amenable Groups

Abstract

In this chapter, by a “dynamical system”, we mean a triple (X, G, T), where X is a topological space, G a group, and $T :G \times X \rightarrow X$ a continuous action of G on X (see Sect. 10.1).

Appendices

Notes

Mean topological dimension for actions of amenable groups was introduced by Gromov in [44]. Its properties were investigated in depth for $\mathbb {Z}$-actions by Lindenstrauss and Weiss in [74]. The exposition in the present chapter closely follows that in [24]. One can define mean topological dimension for actions of uncountable amenable groups by replacing Følner sequences by Følner nets (see the Notes on Chap. 9).

The notion of mean topological dimension has been extended to continuous actions of countable sofic groups by Li [68]. Sofic groups were introduced by Gromov [43] and Weiss [115]. The class of sofic groups is a very vast one. It is known to include in particular all residually finite groups and all amenable groups. Actually, the question of the existence of a non-sofic group is still open. For an introduction to the theory of sofic groups, the reader is referred to the survey paper [87] and to [22, Chap. 7].

Theorem 10.8.1 was obtained by Krieger and the author in [24]. Every residually finite countably-infinite amenable group, and hence every infinite finitely generated linear group (see the Notes on Chap. 9), satisfies the hypotheses of Theorem 10.8.1 (see Exercise 10.8). However, there exist countably-infinite amenable groups, such as the infinite finitely generated amenable simple groups exhibited in [52], that do not satisfy the hypotheses of Theorem 10.8.1. It might be interesting to know whether the conclusion of this theorem remains valid for such groups.

There is an impressive literature dealing with shifts and subshifts over $G = \mathbb {Z}^d$ (see for example the survey papers [69, 71] as well as the references therein). For $d \ge 2$, the study of subshifts of finite type over $\mathbb {Z}^d$ has connections with undecidability questions for tilings of Euclidean spaces.

In his Ph.D. thesis [51, Corollary 4.2.1], Jaworski proved that if G is an abelian group and X is a compact metrizable space with $\dim (X) < \infty$, then every minimal dynamical system (X, G, T) can be embedded in the G-shift on $\mathbb {R}^G$ (see Exercise 10.8). On the other hand, Krieger [62] has shown that if P is a polyhedron and G is a countably-infinite amenable group, then there exist minimal subshifts $X \subset P^G$ whose mean topological dimension is arbitrarily close to $\dim (P)$. It follows in particular from Krieger’s result that there exist minimal dynamical systems (X, G, T), where X is compact and metrizable, that do not embed in the G-shift on $\mathbb {R}^G$.

Exercises

 

• 10.1 Let G be a group and K a topological space. The set $K^G = \prod _{g \in G} K$ is equipped with the product topology. For $g \in G$, denote by $L_g$ the left-multiplication by g, i.e., the map $L_g :G \rightarrow G$ defined by $L_g(h) = gh$ for all $h \in G$. Consider the map $\widetilde{\Sigma } :G \times K^G \rightarrow K^G$ defined by

  $$ \widetilde{\Sigma }(x) = x \circ L_{g^{-1}} $$

  for all $g \in G$ and $x \in K^G$. Show that $\widetilde{\Sigma }$ is a continuous action of G on $K^G$ and that the dynamical systems $(K^G,G,\Sigma )$ and $(K^G,G,\widetilde{\Sigma })$ are topologically conjugate. Hint: use the map $f :K^G \rightarrow K^G$ defined by $f(x)(g) := x(g^{-1})$ for all $x \in K^G$ and $g \in G$.

• 10.2 Let X be a compact metrizable space equipped with a continuous action $T :G \times X \rightarrow X$ of a group G. Let S(X) denote the set consisting of all continuous maps $f :X \rightarrow X$. We equip S(X) with the topology of uniform convergence (i.e., the topology associated with the metric $\rho$ given by $\rho (f_1,f_2) := \sup _{x \in X} d(f_1(x),f_2(x))$ for all $f_1,f_2 \in S(X)$, where d is a metric on X that is compatible with the topology). Consider the map $U :G \times S(X) \rightarrow S(X)$ given by $U(g,f) := T(g,f(x))$ for all $g \in G$, $f \in S(x)$, and $x \in X$.

  • (a) Show that U is a continuous action of G on S(X).

  • (b) Show that the system (X, G, T) embeds in the system (S(X), G, U). Hint: consider the map $\iota :X \rightarrow S(x)$ that sends each $a \in X$ to the constant map $\iota (a) \in S(X)$ defined by $\iota (a)(x) := a$ for all $x \in X$.

• 10.3 (cf. Exercise 4.11). Let (X, d) be a compact metric space equipped with a continuous action $T :G \times X \rightarrow X$ of a countable amenable group G. We associate to each non-empty finite subset $A \subset G$ the metric $d_A$ on X defined by $d_A(x,y) := \max _{g \in A} d(T_g(x),T_g(y))$.

  • (a) Let $\varepsilon > 0$ and let $(F_n)_{n \ge 1}$ be a Følner sequence for G. Show that the limit

    $$ {{\mathrm{mWidim}}}_\varepsilon (X,d,G,T) := \lim _{n \rightarrow \infty } \frac{{{\mathrm{Widim}}}_\varepsilon (X,d_{F_n})}{\vert F_n \vert } $$

    exists, is finite, and does not depend on the choice of the Følner sequence $(F_n)_{n \ge 1}$.

  • (b) Show that

    $$ {{\mathrm{mdim}}}(X,G,T) = \lim _{\varepsilon \rightarrow 0} {{\mathrm{mWidim}}}_\varepsilon (X,d,G,T). $$

• 10.4 (cf. Exercise 7.2). Let G be a group and let X be a topological space. One says that a continuous action T of G on X is topologically mixing if, given any pair $U,V \subset X$ of non-empty open subsets of X, the set of $g \in G$ such that $T_g(U) \cap V = \varnothing$ is finite. Let K be a topological space. Show that the G-shift on $K^G$ is topologically mixing.

• 10.5 Let G be a group. Let K and L be topological spaces. Show that the G-shift on $L^G$ embeds in the G-shift on $K^G$ if and only if the space L embeds in K.

• 10.6 Let X be a non-empty topological space equipped with a continuous action $T :G \times X \rightarrow X$ of a countable amenable group G.

  • (a) Let $\alpha$ be a finite open cover of X. Show that the map $h :\mathcal {P}_{fin}(G) \rightarrow \mathbb {R}$ defined by $h(A) = \log N(\alpha _A)$ (where $\alpha _A$ is defined by (10.2.1) and $N(\cdot )$ is defined in Exercise 6.11) is right-invariant and subadditive.

  • (b) Let $(F_n)_{n \ge 1}$ be a Følner sequence for G. Show that the limit

    $$\begin{aligned} h_{top}(\alpha ,X,G,T) := \lim _{n \rightarrow \infty } \frac{\log N(\alpha _{F_n})}{\vert F_n \vert }, \end{aligned}$$

    exists, is finite, and does not depend on the choice of the Følner sequence $(F_n)$. The quantity $0 \le h_{top}(X,G,T) \le \infty$ defined by

    $$ h_{top}(X,G,T) := \sup _\alpha h_{top}(\alpha ,X,G,T), $$

    where $\alpha$ runs over all finite open covers of X, is called the topological entropy of the dynamical system (X, G, T).

  • (c) Let Y be a topological space equipped with a continuous action $S :G \times Y \rightarrow Y$ of G. Suppose that there exists a surjective continuous map $f :Y \rightarrow X$ such that $f \circ S_g = T_g \circ f$ for all $g \in G$. Show that one has $h_{top}(X,G,T) \le h_{top}(Y,G,S)$.

  • (d) Let $\varphi :X \rightarrow X$ be a homeomorphism of X. Show that the dynamical system $(X,\mathbb {Z},T)$ generated by $\varphi$ satisfies $h_{top}(X,\mathbb {Z},T) = h_{top}(X,\varphi )$, where $h_{top}(X,\varphi )$ is the topological entropy of $(X,\varphi )$ (cf. Exercise 6.11).

• 10.7 Let K be a finite discrete topological space with cardinality k and let G be a countable amenable group. Given $A \in \mathcal {P}_{fin}(G)$, let $\pi _A :K^G \rightarrow K^A$ denote the restriction map. Let $X \subset K^G$ be a non-empty subshift.

  • (a) Let $(F_n)_{n \ge 1}$ be a Følner sequence for G. Show that the limit

    $$ h(X) := \lim _{n \rightarrow \infty } \frac{\log |\pi _{F_n}(X)|}{|F_n|} $$

    exists, is finite, does not depend on the choice of the Følner sequence $(F_n)$, and satisfies $0 \le h(X) \le \log k$. This limit is called the entropy of the subshift X.

  • (b) Show that $h(X) = h_{top}(X,G,\Sigma )$, where $\Sigma$ denotes the G-shift on X and $h_{top}(X,G,\Sigma )$ is the topological entropy of the dynamical system $(X,G,\Sigma )$ (cf. Exercise 10.8).

  • (c) Show that if $Y \subset K^G$ is a subshift such that $X \subset Y$, then $h(X) \le h(Y)$.

• 10.8 One says that a group G is residually finite if the intersection of all the finite index subgroups of G is reduced to the identity element (cf. Exercise 2.5).

  • (a) Let G be a group. Show that the following conditions are equivalent: (1) G is residually finite; (2) the intersection of all the finite index normal subgroups of G is reduced to the identity element; (3) for every element $g \not = 1_G$ in G, there exist a finite group F and a homomorphism $\phi :G \rightarrow F$ such that $\phi (g) \not = 1_F$; (4) for any two distinct elements $g_1,g_2 \in G$, there exist a finite group F and a homomorphism $\phi :G \rightarrow F$ such that $\phi (g_1) \not = \phi (g_2)$; (5) for every finite subset $\Omega \subset G$, there exist a finite group F and a homomorphism $\phi :G \rightarrow F$ whose restriction to $\Omega$ is injective.

  • (b) Show that every finite group is residually finite.

  • (c) Show that every finitely generated abelian group is residually finite.

  • (d) Show that the additive group $\mathbb {Q}$ of rational numbers is not residually finite.

  • (e) Prove that if G is an infinite residually finite group, then G contains subgroups of arbitrarily large finite index. (This shows in particular that every countably-infinite residually finite amenable group satisfies the hypotheses of Theorem 10.8.1.)

  • (f) Show that if G is a residually finite group and K is a topological space, then the periodic points in $K^G$ (i.e., the points whose orbit under the G-shift is finite) are dense in $K^G$.

  • (g) Let G be a group. Show that if there exists a Hausdorff topological space K with more than one point such that the periodic points are dense in $K^G$, then G is residually finite.

• 10.9 Show that every infinite, finitely generated, virtually solvable group satisfies the hypotheses of Theorem 10.8.1. Hint: prove that every infinite, finitely generated, solvable group G has subgroups of arbitrarily large finite index by induction on the solvability degree of G.

• 10.10 Let X be a topological space equipped with a continuous action $T :G \times X \rightarrow X$ of a group G. Suppose that K is a topological space such that X embeds in K. Show that the dynamical system (X, G, T) embeds in the G-shift on $K^G$.

• 10.11 Let X be a compact metrizable space equipped with a continuous action $T :G \times X \rightarrow X$ of a group G. Suppose that $\dim (X) < \infty$. Show that there exists an integer $n \ge 1$ such that the dynamical system (X, G, T) embeds in the shift $((\mathbb {R}^n)^G,G,\Sigma )$.

• 10.12 Let X be a compact space equipped with a continuous action $T :G \times X \rightarrow X$ of a group G. Let K be a Hausdorff space. Show that the following conditions are equivalent: (1) the dynamical system (X, G, T) embeds in the shift $(K^G,G,\Sigma )$; (2) there exists a continuous map $f :X \rightarrow K$ such that, given any two distinct points x and y in X, there is an element $g \in G$ satisfying $f(T_g(x)) \not = f(T_g(y))$.

• 10.13 Let G be a group. One says that an action $T :G \times X \rightarrow X$ of G on a set X is free if $T_g(x) \not = x$ for all $g \in G {\setminus } \{1_G\}$ and $x \in X$. Show that if K is a topological space having more than one point, then the G-shift on $K^G$ is a free action.

• 10.14 (An embedding theorem for free actions [51, Theorem 4.2]). Let G be an infinite group and let X be a compact metrizable space equipped with a continuous action $T :G \times X \rightarrow X$ of G. We suppose that the action of G on X is free (cf. Exercise 10.13) and that X has finite topological dimension $\dim (X) < \infty$.

  • (a) Let x and y be distinct points in X. Show that for every integer $m \ge 1$, there exist elements $g_1,\ldots ,g_m \in G$ such that the points

    $$ T_{g_1}(x),\ldots ,T_{g_m}(x),T_{g_1}(y),\ldots ,T_{g_m}(y) $$

    are pairwise distinct.

  • (b) Show that (X, G, T) embeds in the G-shift on $\mathbb {R}^G$ by using the result of Exercise 10.12 and following the lines of the proof of Theorem 8.3.1.

• 10.15 (An embedding theorem for minimal actions of abelian groups [51, Corollary 4.2.1]). Let G be an abelian group and let X be a compact metrizable space equipped with a continuous action $T :G \times X \rightarrow X$ of G. We suppose that the action of G on X is minimal (i.e., every orbit is dense in X) and that X has finite topological dimension $\dim (X) < \infty$.

  • (a) Let $x \in X$. Show that $H := \{g \in G \vert \; T_g(x) = x\}$ is a subgroup of G and that H does not depend on the choice of the point $x \in X$.

  • (b) Suppose that the group G / H is finite. Show that X is finite and then deduce from the result of Exercise 10.10 that the dynamical system (X, G, T) embeds in the G-shift on $\mathbb {R}^G$.

  • (c) Suppose now that the quotient group $Q := G{/}H$ is infinite. Observe that T induces a continuous free action of Q on X. Then conclude that (X, G, T) embeds in the G-shift on $\mathbb {R}^G$ by using the result of Exercise 10.14 and the canonical embedding $\mathbb {R}^Q \hookrightarrow \mathbb {R}^G$.