### Some uniformly bounded boundary representations of hyperbolic groups

Analysis and Geometry Seminar

2nd November 2023, 3:30 pm – 4:30 pm

Fry Building, 2.04

We prove that some of the boundary representations of (Gromov) hyperbolic groups are uniformly bounded.

More concretely: Suppose *G* is a hyperbolic group, acting geometrically on a (strongly) hyperbolic space *X*. For this talk, "boundary representations" are linear representations *π _{z}* of

*G*coming from the action of

*G*on the Gromov boundary

*Z*of

*X*. These are parametrised by a complex parameter

*z*with 0 < Re(

*z*) < 1. For

*z*=1/2,

*π*is the (unitary) quasi-regular representation on

_{z}*L*. For Re(

^{2}(Z)*z*)≠1/2, there is no obvious unitary structure for

*π*.

_{z}Denote by

*D*the conformal dimension of

*Z*. For 1/2 - 1/

*D*< Re(

*z*) < 1/2 + 1/

*D*, we construct function (Hilbert) spaces on the boundary on which

*π*become uniformly bounded.

_{z}This is joint work with Kevin Boucher.

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