# Events for 06/17/2020 from all calendars

## Groups and Dynamics Seminar

Time: 12:00PM - 1:00PM

Location: 940 9667 3668

Speaker: Bogdan Stankov, ENS, Paris

Title: Non-triviality of the Poisson boundary of certain random walks with finite first moment

Abstract: One equivalent characterization of amenability is the existence of a non-degenerate measure with trivial Poisson boundary (Furstenberg, Rosenblatt, Kaimanovich-Vershik). Vadim Kaimanovich has shown that the Poisson boundary of random walks of finitely supported strictly non-degenerate measures on Thompson's group $F$ is not trivial. It is still not known if $F$ is amenable, which is a famous open question. He asked whether the same statement is true for measures with finite first moment. In this talk we answer that in the positive. More generally, we give a criterion for the non-triviality of the Poisson boundary of random walks on subgroups of groups of piecewise projective homeomorphisms. Furthermore, the simple random walk on the Schreier graph of $F$ has been studied by Mishchenko, who gives another proof of boundary non-triviality. We will show a criterion for non-triviality of an induced random walk on a Schreier graph.

## Noncommutative Geometry Seminar

Time: 1:00PM - 2:00PM

Location: Zoom 942810031

Speaker: Shintaro Nishikawa, Penn State University

Title: Sp(n,1) admits a proper 1-cocycle for a uniformly bounded representation

Abstract: We show that the simple rank one Lie group Sp(n ,1) for any n admits a proper 1-cocycle for a uniformly bounded Hilbert space representation: i.e. it admits a metrically proper affine action on a Hilbert space whose linear part is a uniformly bounded representation. Our construction is a simple modification of the one given by Pierre Julg but crucially uses results on uniformly bounded representations by Michael Cowling. An interesting new feature is that the properness of these cocycles follows from the non-continuity of a critical case of the Sobolev embedding. This work is inspired from Pierre Julg's work on the Baum-Connes conjecture for Sp(n,1).