Given a group G acting properly by isometries 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. We are interested in the following question: when do H and G have the same exponential growth rate ?

This problem has both a combinatorial and a geometric origin. For the combinatorial part, Grigorchuk 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 in F). 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 Grigorchuk 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.

In this talk we are interested in the case where G acts on an arbitrary Gromov hyperbolic space and propose a framework that encompasses both the combinatorial and the geometric point of view. We will see that as soon as the action of G on X is "reasonable" (proper co-compact, cuspidal with parabolic gap, or more generally strongly positively recurrent), then G and H have the same growth rate if and only if H is co-amenable in G. Our strategy is based on a new kind of Patterson-Sullivan measures taking values in a space of bounded operators.

This is a joint work with R. Dougall, B. Schapira and S. Tapie