Boundary representations of locally compact hyperbolic groups

Abstract We develop the theory of Patterson–Sullivan measures for locally compact hyperbolic groups. This theory associates to certain left‐invariant metrics on the group measures on its boundary. Next, we establish irreducibility of the resulting (unitary) Koopman representations for second countable, nonelementary, unimodular locally compact hyperbolic groups. Moreover, the isomorphism type of the unitary representation classifies the given metric on the group up to homothety and bounded additive changes. This generalizes a theorem of Garncarek on discrete hyperbolic groups. Along the way, the Patterson–Sullivan measures are shown to be doubly ergodic. These theorems are used to classify Gelfand pairs among locally compact hyperbolic groups and to answer a question of Caprace, Kalantar, and Monod on type I hyperbolic groups in the unimodular case.