Boundary Representations of Locally Compact Hyperbolic Groups

TL;DR

This paper develops a measure-theoretic boundary theory for locally compact hyperbolic groups, classifies their associated representations, and addresses a key question about their structural type in the unimodular case.

Contribution

It extends Patterson-Sullivan measure theory and representation classification from discrete to locally compact hyperbolic groups, generalizing prior results.

Findings

Koopman representations are irreducible for certain hyperbolic groups.

The isomorphism class of these representations classifies the group's metric up to homothety.

The paper answers a question on the type I property of unimodular hyperbolic groups.

Abstract

We develop the theory of Patterson-Sullivan measures on the boundary of a locally compact hyperbolic group, associating to certain left invariant metrics on the group measures on the boundary. We later prove that for second countable, non-elementary, unimodular locally compact hyperbolic groups the associated Koopman representations are irreducible and their isomorphism type classifies the metric on the group up to homothety and bounded additive changes, generalizing a theorem of Garncarek on discrete hyperbolic groups. We use this to answer a question of Caprace, Kalantar and Monod on type I hyperbolic groups in the unimodular case.