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

Analysis and Geometry Seminar

12th December 2019, 2:00 pm – 3:00 pm

Fry Building, 2.04

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".

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