Soficity, amenability, and dynamical entropy

Abstract

In a previous paper the authors developed an operator-algebraic approach to Lewis Bowen's sofic measure entropy that yields invariants for actions of countable sofic groups by homeomorphisms on a compact metrizable space and by measure-preserving transformations on a standard probability space. We show here that these measure and topological entropy invariants both coincide with their classical counterparts when the acting group is amenable.

Variational principles for topological entropies of subsets

“…[28,31]). Recently, Kerr and Li obtained a variational principle of topological entropy and measure-theoretic entropy for actions of sofic group [19,20].…”

Soficity and variations on Higman’s group

“…The remaining implication (2) ⇒ (3) is significantly more complicated and uses that for an amenable group, any two approximations into the same Sym(n) are almost conjugate. This result is due to Helfgott and Juschenko in the form given but, as they explain, has origins in Elek and Szabo [9], builds on a lemma from Kerr and Li [19], and is also comparable to Arzhantseva and Pǎunescu [1]. Helfgott and Juschenko's proof is a delicate analysis of the interplay between sofic approximations and the Følner characterization of amenability.…”

“…The additional assumption that the group G is amenable gives better control of the sofic approximations. The key result is a theorem which is due to Helfgott and Juschenko [16] in the form we will use and has origins in Elek and Szabo [9] and Kerr and Li [19]. It spells out a manner in which any two sofic approximations of an amenable group are almost conjugate.…”

Positive entropy actions of countable groups factor onto Bernoulli shifts

“…We remind the reader that the class of sofic groups contains the countable amenable groups, and it is an open problem whether every countable group is sofic. Sofic entropy is an extension of Kolmogorov-Sinai entropy, as when the acting group is amenable the two notions coincide [5,31]. For sofic groups G, the Bernoulli shift G (L G , λ G ) has sofic entropy H(L, λ) as expected [4,33], thus implying the non-isomorphism of many Bernoulli shifts.…”

“…Work of Bowen [4], combined with improvements by Kerr and Li [30,31], created the notion of sofic entropy for p.m.p. actions of sofic groups.…”