Growth gap in hyperbolic groups and amenability

TL;DR

This paper establishes a connection between subgroup amenability and exponential growth rates in hyperbolic groups acting on hyperbolic spaces, revealing a growth gap for groups with property (T).

Contribution

It generalizes the amenability conjecture to hyperbolic groups acting on CAT(-1)-spaces and introduces a spectral radius criterion for subgroup amenability.

Findings

H is co-amenable in G iff their exponential growth rates match.

Existence of a growth gap for groups with property (T).

Extension of Corlette's theorem to broader hyperbolic contexts.

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.