Activities This Week

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.

A group is said to have a fixed-point property with respect to some class of metric spaces if any isometric action of the group on any space in the class admits a fixed point.

In this talk, I will focus on fixed-point properties with respect to (classes of) Banach spaces. I will survey some results regarding groups with and without these fixed-point properties and then present a recent result of mine regarding fix-point properties for random groups with respect to l^p spaces.

Jacobians of CM curves are abelian varieties with a particularly large endomorphism algebra, which provides them with a rich arithmetic structure. The motivating question for the results in this talk is whether we can find hyperelliptic and non-hyperelliptic curves with maximal CM by a given order whose Jacobians are isogenous. Joint work with Sorina Ionica, and Jeroen Sijsling considers this question in genus 3 by using the catalogue of CM fields in the LMFDB, and found a (small) list of such isogenous Jacobians. This talk describes the main constructions, some results, and Shimura class groups.