Cost, ℓ2-Betti numbers and the sofic entropy of some algebraic actions

Abstract

In 1987, Ornstein and Weiss discovered that the Bernoulli 2-shift over the rank two free group factors onto the seemingly larger Bernoulli 4-shift. With the recent creation of an entropy theory for actions of sofic groups (in particular free groups), their example shows the surprising fact that entropy can increase under factor maps. In order to better understand this phenomenon, we study a natural generalization of the Ornstein-Weiss map for countable groups. We relate the increase in entropy to the cost and to the first ℓ 2 -Betti number of the group. More generally, we study coboundary maps arising from simplicial actions and, under certain assumptions, relate ℓ 2 -Betti numbers to the failure of the Yuzvinsky addition formula. This work is built upon a study of entropy theory for algebraic actions. We prove that for actions on profinite groups via continuous group automorphisms, topological sofic entropy is equal to measure sofic entropy with respect to Haar measure whenever the homoclinic subgroup is dense. For algebraic actions of residually finite groups we find sufficient conditions for the sofic entropy to be equal to the supremum exponential growth rate of periodic points.

Examples in the entropy theory of countable group actions

“…This is a closed -invariant subgroup. So K /K is an algebraic action (called an Ornstein-Weiss factor in [GS15] where bounds on its entropy are obtained). In [Pop06a], it is shown that the cohomology group of K /K (with respect to Haar measure on K /K ) is isomorphic to Hom( , R/Z) × Hom(K , R/Z).…”

Local and doubly empirical convergence and the entropy of algebraic actions of sofic groups

“…X is a profinite group and G X has a dense homoclinic group, and doubly empirical convergence model-measure sofic entropy up to date with that of the entropy defined by [16], [4]. Moreover we are able to do this in a relatively painless manner without having to essentially reproduce the work in [13], [9], [16], [7], [5]. For example, as a consequence of Corollary 1.4, our previous results connecting Fuglede-Kadison determinants and sofic measure entropy (namely Theorem 1.1 of [13]) now hold…”

“…Gaboriau-Seward in [9], who gave the first general results on equality of topological and measure-theoretic entropy without directly computing either entropy. Specifically, they showed equality of the entropies when G X by automorphisms, where X is any profinite group and when the homoclinic group of G X is dense.…”

“…, with action given by (1) and with λ(f ) injective (implicit in [13]), (iv) X is a profinite group, µ = m X , where G X by automorphisms, and G X has a dense homoclinic group (implicit in [9]).…”

Entropy on modules over the group ring of a sofic group

“…In this paper, we give a partial generalization of Peters' formula to the case of sofic group actions. The main inputs of the proof are an approximation formula of topological entropy given in [7,Lemma 4.8] and some techniques appeared in [12].…”