**Abstract:**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. 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. This talk will review some recent history of this problem involving transfer operators for skew product extensions; and present our new approach which yields 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". This is joint work with R. Coulon, B. Schapira and S. Tapie.

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Meeting ID: 910 4725 1047

Password: 5w97m2

## Date:

Tue, 02/06/2020 - 14:00 to 15:00