# Shannon's theorem for locally compact groups

## TL;DR

This paper extends classical results on random walks and Poisson boundaries from discrete groups to locally compact groups, providing new criteria and applications for understanding their asymptotic behavior.

## Contribution

It generalizes the Shannon-McMillan-Breiman theorem and boundary identification criteria to locally compact groups, broadening the scope of geometric and probabilistic analysis.

## Key findings

- Generalized Shannon-McMillan-Breiman theorem for locally compact groups
- Extended Kaimanovich's boundary criteria to new classes of groups
- Identified Poisson boundaries for groups acting on nonpositively curved spaces

## 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.