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 \(\mathbb {R}^G\).