Status Confirmed Seminar Series MATH-IHES Subjects hep-th Date Monday 10 December 2018 Time 16:30 Institute IHES Seminar Room Amphithéâtre Léon Motchane Speaker's Last Name Tapie Speaker's First Name Samuel Speaker's Email Address Speaker's Institution Université de Nantes Title Growth gap, amenability and coverings Abstract Let \Gamma; be a group acting by isometries on a proper metric space (X,d). The critical exponent \delta_\Gamma (X) is a number which measures the complexity of this action. The critical exponent of a subgroup \Gamma'<\Gamma; is hence smaller than the critical exponent of \Gamma. When does equality occur? It was shown in the 1980s by Brooks that if X is the real hyperbolic space, \Gamma' is a normal subgroup of \Gamma and \Gamma is convex-cocompact, then equality occurs if and only if \Gamma/\Gamma' is amenable. At the same time, Cohen and Grigorchuk proved an analogous result when \Gamma is a free group acting on its Cayley graph. When the action of \Gamma on X is not cocompact, showing that the equality of critical exponents is equivalent to the amenability of \Gamma/\Gamma' 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 \Gamma with a proper action on a Gromov hyperbolic space. Joint work with R. Coulon, R. Dougall and B. Schapira. arXiv Preprint Number Comments Séminaire Géométrie et groupes discrets Attachments

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