We prove that some of the boundary representations of (Gromov) hyperbolic groups are uniformly bounded.
One can construct complementary series representations of SL(2,R) from its action on the circle; this work is an attempt to generalise parts of this theory to hyperbolic groups.
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 Hausdorff dimension of Z. For 1/2 - 1/D < Re(z) < 1/2, we construct function spaces on Z on which πz become uniformly bounded.
This is joint work with Kevin Boucher.
