Boundary representations of hyperbolic groups

Abstract

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

v2: added an appendix explaining double ergodicity of Patterson-Sullivan measures in the setting of the paper

References in corpus (3)

• Asymptotic entropy and green speed for random walks on countable groups
• Analogs of principal series representations for Thompson's groups $F$ and $T$
• On irreducibility of Koopman representations of Higman-Thompson groups

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