We prove a general version of the amenability conjecture in the unified setting of a Gromov hyperbolic group G acting properly cocompactly either on its Cayley graph, or on a CAT(-1)-space. Namely, for any subgroup H of G, we show that H is co-amenable in G if and only if their exponential growth rates (with respect to the prescribed action) coincide. For this, we prove a quantified, representation-theoretical version of Stadlbauer’s amenability criterion for group extensions of a topologically transitive subshift of finite type, in terms of the spectral radii of the classical Ruelle transfer operator and its corresponding extension. As a consequence, we are able to show that, in our enlarged context, there is a gap between the exponential growth rate of a group with Kazhdan’s property (T) and the ones of its infinite index subgroups. This also generalizes a well-known theorem of Corlette for lattices of the quaternionic hyperbolic space or the Cayley hyperbolic plane.

Growth gap in hyperbolic groups and amenability / Coulon, Rémi; Dal’Bo, Françoise; Sambusetti, Andrea. - In: GEOMETRIC AND FUNCTIONAL ANALYSIS. - ISSN 1016-443X. - STAMPA. - 28:5(2018), pp. 1260-1320. [10.1007/s00039-018-0459-6]

Growth gap in hyperbolic groups and amenability

Coulon, Rémi;Dal’Bo, Françoise;Sambusetti, Andrea
2018

Abstract

We prove a general version of the amenability conjecture in the unified setting of a Gromov hyperbolic group G acting properly cocompactly either on its Cayley graph, or on a CAT(-1)-space. Namely, for any subgroup H of G, we show that H is co-amenable in G if and only if their exponential growth rates (with respect to the prescribed action) coincide. For this, we prove a quantified, representation-theoretical version of Stadlbauer’s amenability criterion for group extensions of a topologically transitive subshift of finite type, in terms of the spectral radii of the classical Ruelle transfer operator and its corresponding extension. As a consequence, we are able to show that, in our enlarged context, there is a gap between the exponential growth rate of a group with Kazhdan’s property (T) and the ones of its infinite index subgroups. This also generalizes a well-known theorem of Corlette for lattices of the quaternionic hyperbolic space or the Cayley hyperbolic plane.
2018
analysis; geometry and topology
01 Pubblicazione su rivista::01a Articolo in rivista
Growth gap in hyperbolic groups and amenability / Coulon, Rémi; Dal’Bo, Françoise; Sambusetti, Andrea. - In: GEOMETRIC AND FUNCTIONAL ANALYSIS. - ISSN 1016-443X. - STAMPA. - 28:5(2018), pp. 1260-1320. [10.1007/s00039-018-0459-6]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1182393
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