Title: Shannon’s theorem and Poisson boundaries for locally compact groups
Speaker: Giulio Tiozzo
Speaker Info: University of Toronto
Brief Description: Dynamics Day Lecture
The Furstenberg-Poisson boundary is a canonical measurable object associated to a random walk on a group, and a basic question in the field is its identification with a topological boundary coming from some geometric structure on the group.Date: Tuesday, March 03, 2020
For discrete groups, Avez, Derriennic, Vershik, and in particular Kaimanovich formulated geometric criteria for the identification of the Poisson boundary, based on the approximation of the random walk by geodesics in the space.
We obtain a version of these criteria for locally compact groups, settling a long-standing conjecture by Kaimanovich. The crucial step is a proof of the Shannon-McMillan-Breiman theorem, which addresses a question of Derriennic.
This has applications to isometries of hyperbolic and CAT(0) spaces, as well as Diestel-Leader graphs and Sol manifolds. Joint with B. Forghani.