# Growth gap in hyperbolic groups and amenability

@article{Coulon2017GrowthGI,
title={Growth gap in hyperbolic groups and amenability},
author={R{\'e}mi Coulon and Françoise Dal'bo and Andrea Sambusetti},
journal={Geometric and Functional Analysis},
year={2017},
volume={28},
pages={1260-1320}
}
• Published 21 September 2017
• Mathematics
• Geometric and Functional Analysis
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…
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