##### Department of Mathematics,

University of California San Diego

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### Math 211B - Group Actions Seminar

## Elad Sayag

#### Tel Aviv University

## Entropy, ultralimits and Poisson boundaries

##### Abstract:

In many important actions of groups there are no invariant measures. For example: the action of a free group on its boundary and the action of any discrete infinite group on itself. The problem we will discuss in this talk is 'On a given action, how invariant measure can be? '. Our measuring of non-invariance will be based on entropy (f-divergence).

In the talk I will describe the solution of this problem for the Free group acting on its boundary and on itself. For doing so we will introduce the notion of ultra-limit of $G$-spaces, and give a new description of the Poisson-Furstenberg boundary of $(G,k)$ as an ultra-limit of $G$ action on itself, with 'Abel sum' measures. Another application will be that amenable groups possess KL-almost-invariant measures (KL stands for the Kullback-Leibler divergence). All relevant notions, including the notion of Poisson-Furstenberg boundary and the notion of Ultra-filters will be explained during the talk. This is a master thesis work under the supervision of Yehuda Shalom.

Host: Brandon Seward

### October 27, 2022

### 10:00 AM

Zoom ID 967 4109 3409

(email an organizer for the password)

Research Areas

Ergodic Theory and Dynamical Systems****************************