3:00 pm Wednesday, April 15, 2015
Group Actions & Dynamics : Polish models and sofic entropy by Benjamin Hayes (Vanderbilt) in RLM 9.166
Let G be a countable discrete with a pmp action on a standard probabilty space (X,m). A topological model for this action, is a separable, metrizable topological space X', with an action of G on X by homeomorphisms and a Borel G-invariant probablity measure m' on X so that as (X,m) is isomorphic to (X',m') as pmp actions of G. It is know that compact models always exist. When G is sofic, Kerr-Li showed how one can compute the entropy (as originally defined by Lewis Bowen) of a pmp action when one is given a compact model for the action. We show how to do this, when one is merely given a Polish model. Despite the fact that compact models always exist, this generality turns out to be useful. For example, we deduce consequences for the Koopman representation of a pmp action under positive entropy assumptions. These results appear to be the first of their kind in the nonamenable case. We also compute the entropy for Gaussian actions .Time permitting, we will comment on the techniques, which are mostly ``linear" relying on facts about representations of C*-algebras. No knowledge of sofic groups or sofic entropy will be assumed.
Submitted by