Measure conjugacy invariants for actions of countable sofic groups

Abstract

Sofic groups were defined implicitly by Gromov and explicitly by Weiss. All residually finite groups (and hence all linear groups) are sofic. The purpose of this paper is to introduce, for every countable sofic group G G , a family of measure-conjugacy invariants for measure-preserving G G -actions on probability spaces. These invariants generalize Kolmogorov-Sinai entropy for actions of amenable groups. They are computed exactly for Bernoulli shifts over G G , leading to a complete classification of Bernoulli systems up to measure-conjugacy for many groups, including all countable linear groups. Recent rigidity results of Y. Kida and S. Popa are utilized to classify Bernoulli shifts over mapping class groups and property (T) groups up to orbit equivalence and von Neumann equivalence, respectively.

A Juzvinskii addition theorem for finitely generated free group actions

“…(Later a more general theory of sofic entropy was introduced in [Bo10b], on which we have little to L. Bowen and Y. Gutman say in the present paper.) In [Bo10c], a proof is claimed that the above addition formula extends to the case where is a finitely generated free group, the entropy is replaced with the f -invariant, and G is either totally disconnected, a Lie group, or a connected abelian group (whenever the f -invariant is well defined).…”

Universal Borel actions of countable groups

“…If A i is a null set, then we define µ(A i ) log(µ(A i )) = 0. Finally we will make use of the following result, which is implicitly contained in Bowen [2]. Since Bowen does not state this result explicitly, we will briefly explain how to deduce Theorem 6.11 from the results in [2].…”

“…In Section 5, we will prove that if n is a sufficiently large odd integer and G = B(2, n) is the free 2-generator Burnside group of exponent n, then E(G, 2) is not essentially free. Finally, in Section 6, we will switch our attention from universal actions to G-universal actions; and we will use the remarkable recent work of Bowen [2] on the ergodic theory of sofic groups to prove Theorem 1.7.…”

Weak isomorphisms between Bernoulli shifts

“…From this extension and Stepin's result it follows that Bernoulli shifts over free groups are completely classified up to measure-conjugacy by base-measure entropy. In [Bo08b], the author extended these results to sofic groups, a class of groups that contains all residually finite groups.…”