Status  Confirmed 
Seminar Series  MATHIHES 
Subjects  hepth 
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 convexcocompact, 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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