Ultralimits, Amenable actions and Entropy

TL;DR

This paper investigates the minimal Furstenberg entropy for amenable group actions, showing it matches the group's action on itself, and computes this minimal entropy for free and hyperbolic groups using ultralimit techniques.

Contribution

It introduces an ultralimit approach to analyze Furstenberg entropy, extending previous work and providing explicit entropy values for boundary actions of free and hyperbolic groups.

Findings

Minimal Furstenberg entropy for amenable actions equals that of the group acting on itself.

Computed minimal entropy for free group boundary actions.

Extended ultralimit techniques to analyze Poisson boundaries.

Abstract

In this paper we show that the minimal value of Furstenberg entropy (along all measures, not restricting to stationary ones) for any amenable action is the same as for the action of the group on itself. Using the boundary amenability result of Adams, this allows us to compute the minimal value of the entropy over all the measure classes in the boundary of the free group. Similar results are proved for the action of a hyperbolic group on its Gromov boundary. Our main tool is an ultralimit realization of the Poisson boundary of a time dependent matrix-valued random walk on the group. This extends and refines the results and tools of previous paper of the author with Y. Shalom.