Séminaire Géométrie et groupes discrets

Growth gap, amenability and coverings

by Prof. Samuel Tapie (Université de Nantes)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette
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.
Organized by

Fanny Kassel