Buondì,

Segnalo un seminario di geometria per domani pomeriggio. A presto, Bruno

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06-Dec-2018 - 16:00 Sala Seminari (Dip. Matematica) http://www.dm.unipi.it/webnew/it/aula/sala-seminari-dip-matematica The amenability conjecture for Gromov-hyperbolic groups http://www.dm.unipi.it/webnew/it/seminari/amenability-conjecture-gromov-hyperbolic-groups Andrea Sambusetti

Abstract:

We will present a classical conjecture on amenability for groups related to growth. The original result, which is due to Cohen and, independently, Grigorchuk, dates back to the 80's: a group Q finitely presented as F/N, where F is a free group, is amenable if and only if the exponential growth rates (or _entropies)_ \omega(N) and \omega(F) of N and F coincide.

This is related to Kesten's criterion for amenability, and has been declined in many different ways and different contexts during the years. The most accredited version of the Amenability Problem in the last decade was: given a group G acting properly on a CAT(-1) space X and a normal subgroup N of G, does the equality \omega(N) = \omega(G) imply that the quotient group Q = G/N is amenable? We prove a general version of the amenability conjecture for Gromov word-hyperbolic groups or a cocompact groups of isometries of a CAT(-1)-space. For this, we prove a quantified version of an amenability criterion due to Stadlbauer (generalizing Kesten's 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 entropy of a group with Kazhdan’s property (T) and the entropies of all 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. We will try to sketch the overall strategy and give an idea of all the background material needed in the proof (dynamics on hyperbolic groups, Gromov's coding, transfer operator for subshifts of finite type, spectral amenability criterions for group extensions etc). The seminar is based on the joint work with F. Dal'Bo and R. Coulon https://arxiv.org/pdf/1709.07287.pdf [1]