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Pré-Publication, Document De Travail Année : 2017

Growth gap in hyperbolic groups and amenability

Résumé

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.
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Dates et versions

hal-01591471 , version 1 (21-09-2017)
hal-01591471 , version 2 (09-10-2017)
hal-01591471 , version 3 (23-08-2018)

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Rémi Coulon, Françoise Dal'Bo-Milonet, Andrea Sambusetti. Growth gap in hyperbolic groups and amenability. 2017. ⟨hal-01591471v1⟩
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