Rhiannon Dougall

Measures on the boundary: an application to critical exponents of subgroups of groups acting on Gromov hyperbolic spaces, and amenability

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

One of the first things we learn about a (proper) Gromov hyperbolic geodesic space X is the construction of the visual boundary ∂X. An ergodic theorist then learns that for a non-elementary discrete group of isometries G acting properly on X, there is an interesting family of s-quasi-conformal measures on ∂X, called Patterson-Sullivan measures. And the smallest such s coincides with the critical exponent δ_G of G: that is, δ_G is the exponential growth in R of Gx ∩ B(x,R) ⊂ X. A general question might be, given a subgroup H of G, how do we compare δ_H and δ_G? For G with "good dynamics", we expect that δ_G = δ_H if and only if H is co-amenable in G. Indeed, in the case that X is a rank 1 symmetric space, some results were known using Brooks' characterisation of amenability in terms of the bottom of the spectrum of the Laplacian. I will motivate the question, and say something about our new approach, for which we have an optimal conclusion, that if the action of G is SPR, then we have δ_G = δ_H if and only if H is co-amenable in G. What is particularly appealing about our method is the construction of a new "twisted Patterson-Sullivan measure".