Polish Models and Sofic Entropy

Abstract

For actions of a sofic group on probability spaces, the entropy has been defined by Bowen, with an extension by Kerr-Li. In particular, when the action is by homeomorphisms of a compact space preserving a given measure, Kerr-Li show one can compute the measure-theoretic entropy in a manner which uses the topology of the space. We show how to compute the entropy of a an action of homeomorphisms of a Polish space preserving a given measure, also in a manner which uses the topology of the space. We give applications to spectral properties of actions with positive entropy, as well as for actions of completely positive entropy.

26 pages. Removed the section on distal actions as we will prove more general results in "Mixing and Spectral Gap Relative to Pinsker Factors for Sofic Groups". This is the final version to appear in Journal of Institute of Mathematics of Jussieu

References in corpus (5)

• An l^{p}-Version of von-Neumann Dimension For Banach Space Representations of Sofic Groups
• Mixing and Spectral Gap Relative to Pinsker Factors for Sofic Groups
• Sofic groups: graph products and graphs of groups
• Convex Structures Revisited
• Positive sofic entropy implies finite stabilizer

Cited by in corpus (4)

• Fuglede-Kadison Determinants and Sofic Entropy
• Examples in the entropy theory of countable group actions
• Sofic Entropy of Gaussian Actions
• Entropy, products, and bounded orbit equivalence