Le Bois Marie
35, route de Chartres
Let Γ be a group acting by isometries on a proper metric space (X,d). The critical exponent δΓ(X) is a number which measures the complexity of this action. The critical exponent of a subgroup Γ'<Γ is hence smaller than the critical exponent of Γ. When does equality occur? It was shown in the 1980s by Brooks that if X is the real hyperbolic space, Γ' is a normal subgroup of Γ and Γ is convex-cocompact, then equality occurs if and only if Γ/Γ' is amenable. At the same time, Cohen and Grigorchuk proved an analogous result when Γ is a free group acting on its Cayley graph.
When the action of Γ on X is not cocompact, showing that the equality of critical exponents is equivalent to the amenability of Γ/Γ' requires an additional assumption: a "growth gap at infinity". I will explain how, under this (optimal) assumption, we can generalize the result of Brooks to all groups Γ with a proper action on a Gromov hyperbolic space.
Joint work with R. Coulon, R. Dougall and B. Schapira.