Modul: MAT760 Ergodic Theory and Dynamical Systems Seminar

## e-Seminar: Critical exponents via twisted Patterson-Sullivan measures

Talk by Dr. Rhiannon Dougall

**Date:** 27.04.20 **Time:** 13.45 - 14.45 **Room:** Y27H28

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 of 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 \delta_G-quasi-conformal measures on the boundary, called Patterson-Sullivan measures. The parameter \delta_G is called the critical exponent of G, and is equal to the exponential growth rate of the orbit Gx in X. Given a subgroup H of G, how do we compare \delta_H and \delta_G? For G with "good dynamics", we expect that \delta_G = \delta_H if and only if H is co-amenable in G. (As 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.) We obtain the theorem: if the action of G is SPR, then we have \delta_G = \delta_H if and only if H is co-amenable in G. What is particularly appealing about our method is the construction of a "twisted Patterson-Sullivan measure".