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 series of lectures, we will revisit this problem in the general context of a group G acting on a CAT(-1) space, or more generally Gromov hyperbolic space X. We will first introduce measure-theoretic tools to study the dynamics of the geodesic flow on X. Then, given a subgroup H of G, we will explain the construction of a Hilbert-valued measure on the boundary at infinity of X. The “dimension” of this measure will allow us to characterize the equality of the exponential growth rates of H and G, in terms of the algebraic properties of the quotient G/H.