Measure conjugacy invariants for actions of countable sofic groups

By Lewis Bowen. 2008

210 citations recorded by the source corpus.

Sofic groups were defined implicitly by Gromov in [Gr99] and explicitly by Weiss in [We00]. All residually finite groups (and hence every linear group) is sofic. The purpose of this paper is to introduce, for every countable sofic group $G$, a family of measure-conjugacy invariants for measure-preserving $G$-actions on probability spaces. These invariants generalize Kolmogorov-Sinai entropy for actions of amenable groups. They are computed exactly for Bernoulli shifts over $G$, 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.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/61bec943bd0cd29ef079365bc37afbf09b11c982

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Citations in the corpus

Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture (2013, background): “Another proof was obtained Kerr and Li in [75] as an application of the theory of entropy for actions of sofic groups developed by Bowen, Kerr, and Li (see [7], [75], and [76]).”
Sofic measure entropy via finite partitions (2011, methodology): “This provides for a relatively concise computation of the entropy of general Bernoulli actions of sofic groups which builds on arguments from [1].”
ENTROPY, DETERMINANTS, AND L 2 -TORSION (2012, background): “Recently, entropy theory has been extended to actions of countable sofic groups [5], [36], which include all countable amenable groups and countable residually finite groups.”
Soficity, amenability, and dynamical entropy (2010, background): “In [2] Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable discrete sofic group on a standard probability space admitting a generating finite partition.”
Combinatorial Independence and Sofic Entropy (2012, methodology): “By measuring the asymptotic exponential growth of dynamical models which are compatible with a fixed sofic approximation sequence, Bowen defined in [6] a collection of invariants for probability-measurepreserving actions of a countable sofic group admitting a generating partition with finite Shannon”
Entropy and the variational principle for actions of sofic groups (2010, unknown): “We write AP(P, F, δ, σ) for the set of approximating ordered partitions as in [5], and hΣ,μ(P, F, δ), hΣ,μ(P, F ), and h ′ Σ,μ(P), for the entropy quantities in [5].”
Fuglede–Kadison Determinants and Sofic Entropy (2014, methodology): “Seminal work of Bowen in [4] extended the definition of measure entropy for actions of amenable groups to the much larger class of sofic groups assuming the action has a finite generating partition.”
Compact Group Automorphisms, Addition Formulas and Fuglede-Kadison Determinants (2010, background): “Recently, entropy has been defined for continuous actions of a countable sofic group on compact metrizable spaces and measure-preserving actions of a countable sofic group on standard probability measure spaces, with respect to a sofic approximation sequence of the sofic group [4, 34].”