Boundary representations of hyperbolic groups

Abstract

Let $Γ$ be a Gromov hyperbolic group, endowed with an arbitrary left-invariant hyperbolic metric, quasi-isometric to a word metric. The action of $Γ$ on its boundary $\partialΓ$ endowed with the Patterson-Sullivan measure $μ$, after an appropriate normalization, gives rise to a faithful unitary representation of $Γ$ on $L^2(\partialΓ,μ)$. We show that these representations are irreducible, and give criteria for their unitary equivalence in terms of the metrics on $Γ$. Special cases include quasi-regular representations on the Poisson boundary.