Boundary Representations of Locally Compact 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.