Sofic entropy of Gaussian actions

Abstract

Abstract. Associated to any orthogonal representation of a countable discrete group is a probability measure-preserving action called the Gaussian action. Using the Polish model formalism we developed before, we compute the entropy (in the sense of Bowen, Kerr-Li) of Gaussian actions when the group is sofic. Computations of entropy for Gaussian actions has only been done when the acting group is abelian and thus our results are new even in the amenable case. Fundamental to our approach are methods of noncommutative harmonic analysis and C * -algebras which replace the Fourier analysis used in the abelian case.

Examples in the entropy theory of countable group actions

“…On the other hand, if the action is ergodic then there is an equivalent sofic approximation Σ ′ such that Σ ′ is by expanders. In this case, Σ is said to be an ergodic sofic approximation [Hay17b].…”

“…For intuition, if H is finite-dimensional then the Gaussian action is the action of Γ on H with respect to the standard Gaussian measure. Ben Hayes computed the entropy of Gaussian actions in [Hay17b]:…”

Locally compact sofic groups

“…The variational principle holds: topological Σ-entropy is the supremum of h Σ (Γ, X, µ) over all Γinvariant measures µ. Moreover the entropies of Bernoulli shifts, Gaussian actions and many algebraic actions have been computed [Bow10,KL11a,Hay17,Hay16]. These entropies agree with their classical counterparts when Γ is amenable [KL13].…”

The work of Lewis Bowen on the entropy theory of non-amenable group actions

“…Many examples have been the object of study: Markov actions, actions of algebraic origin [15], Gaussian actions [17]. T. Austin and P. Burton [2] have constructed uncountably many actions with the same entropy with completely positive sofic entropy, pairwise not isomorphic (and none of them being a factor of a Bernoulli shift).…”