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]