Boundary Representations of Locally Compact Hyperbolic Groups

2023/09/22 by Michael Glasner, Glasner, Michael
Mathematics · #20F67 #22D10 #37A40 #43A65 #Advanced Algebra and Geometry #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Group Theory (math.GR) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2309.12921

openalex publication_date 2023/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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