# 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} }

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…

## 16 Citations

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