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Growth gap in hyperbolic groups and amenability

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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.

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Authors and Affiliations

  1. Univ Rennes, CNRS, IRMAR - UMR 6625, 35000, Rennes, France

    Rémi Coulon & Françoise Dal’Bo

  2. Dipartimento di Matematica G.Castelnuovo, Sapienza Università di Roma, P.le Aldo Moro 5, 00198, Rome, Italy

    Andrea Sambusetti

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  1. Rémi Coulon
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  2. Françoise Dal’Bo
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  3. Andrea Sambusetti
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Correspondence to Rémi Coulon.

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Coulon, R., Dal’Bo, F. & Sambusetti, A. Growth gap in hyperbolic groups and amenability. Geom. Funct. Anal. 28, 1260–1320 (2018). https://doi.org/10.1007/s00039-018-0459-6

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  • DOI: https://doi.org/10.1007/s00039-018-0459-6

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