Jan Spakula

University of Southampton

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 πz 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, πz is the (unitary) quasi-regular representation on L2(Z). For Re(z)≠1/2, there is no obvious unitary structure for πz.
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 πz become uniformly bounded.
This is joint work with Kevin Boucher.

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