Asymptotic Schur orthogonality in hyperbolic groups with application to monotony

TL;DR

This paper generalizes Schur orthogonality relations for representations of Gromov hyperbolic groups and demonstrates their monotonicity properties, with applications to harmonic measures and random walks.

Contribution

It introduces a generalized Schur orthogonality framework for hyperbolic groups and applies it to establish monotonicity of certain group representations.

Findings

Generalized Schur orthogonality relations for hyperbolic groups

Monotonicity of representations associated with Patterson-Sullivan measures

Applicability to harmonic measures of random walks

Abstract

We prove a generalization of Schur orthogonality relations for certain classes of representations of Gromov hyperbolic groups. We apply the obtained results to show that representations of non-abelian free groups associated to the Patterson-Sullivan measures corresponding to a wide class of invariant metrics on the group are monotonous in the sense introduced by Kuhn and Steger. This in particular includes representations associated to harmonic measures of a wide class of random walks.