(joint work with Françoise Dal'Bo and Andrea Sambusetti)

Given a finitely generated group G acting properly on a metric space X,

the exponential growth rate of G with respect to X measures "how big"

the orbits of G are. If H is a subgroup of G, its exponential growth

rate is bounded above by the one of G. In this work we are interested in

the following question: what can we say if H and G have the same

exponential growth rate? This problem has both a combinatorial and a

geometric origin. For the combinatorial part, Grigorchuck and Cohen

proved in the 80's that a group Q = F/N (written as a quotient of the

free group) is amenable if and only if N and F have the same exponential

growth rate (with respect to the word length). About the same time,

Brooks gave a geometric interpretation of Kesten's amenability criterion

in terms of the bottom of the spectrum of the Laplace operator. He

obtained in this way a statement analogue to the one of Grigorchuck and

Cohen for the deck automorphism group of the cover of certain compact

hyperbolic manifolds. These works initiated many fruitful developments

in geometry, dynamics and group theory. We focus here one the class of

Gromov hyperbolic groups and propose a framework that encompasses both

the combinatorial and the geometric point of view. More precisely we

prove that if G is a hyperbolic group acting properly co-compactly on a

metric space X which is either a Cayley graph of G or a CAT(-1) space,

then the growth rate of H and G coincide if and only if H is co-amenable

in G. In addition if G has Kazhdan property (T) we prove that there is a

gap between the growth rate of G and the one of its infinite index

subgroups.

Given a finitely generated group G acting properly on a metric space X,

the exponential growth rate of G with respect to X measures "how big"

the orbits of G are. If H is a subgroup of G, its exponential growth

rate is bounded above by the one of G. In this work we are interested in

the following question: what can we say if H and G have the same

exponential growth rate? This problem has both a combinatorial and a

geometric origin. For the combinatorial part, Grigorchuck and Cohen

proved in the 80's that a group Q = F/N (written as a quotient of the

free group) is amenable if and only if N and F have the same exponential

growth rate (with respect to the word length). About the same time,

Brooks gave a geometric interpretation of Kesten's amenability criterion

in terms of the bottom of the spectrum of the Laplace operator. He

obtained in this way a statement analogue to the one of Grigorchuck and

Cohen for the deck automorphism group of the cover of certain compact

hyperbolic manifolds. These works initiated many fruitful developments

in geometry, dynamics and group theory. We focus here one the class of

Gromov hyperbolic groups and propose a framework that encompasses both

the combinatorial and the geometric point of view. More precisely we

prove that if G is a hyperbolic group acting properly co-compactly on a

metric space X which is either a Cayley graph of G or a CAT(-1) space,

then the growth rate of H and G coincide if and only if H is co-amenable

in G. In addition if G has Kazhdan property (T) we prove that there is a

gap between the growth rate of G and the one of its infinite index

subgroups.

## Date:

Thu, 02/11/2017 - 10:30 to 11:30

## Location:

hyperbolic groups and amenability