Shannon's theorem for locally compact groups

Abstract

We consider random walks on locally compact groups, extending the geometric criteria for the identification of their Poisson boundary previously known for discrete groups. First, we prove a version of the Shannon-McMillan-Breiman theorem, which we then use to generalize Kaimanovich's ray approximation and strip approximation criteria. We give several applications to identify the Poisson boundary of locally compact groups which act by isometries on nonpositively curved spaces, as well as on Diestel-Leader graphs and horocylic products.

Sublinearly Morse Boundary II: Proper geodesic spaces

“…We will use the ray approximation criterion from [Kai00] for the Poisson boundary (for this precise version, see [FT18]). Theorem 6.1.…”