Groups and Dynamics Seminar

Date: June 17, 2020

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.