| Résume | 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 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_z is the (unitary)
quasi-regular representation on L²(Z). For Re(z)≠1/2, there is no
obvious unitary structure for π_z.
Denote by D the Hausdorff dimension of Z. For 1/2 - 1/D < Re(z) < 1/2 +
1/D, we construct function spaces on the boundary on which
π_z become uniformly bounded.
This is joint work with Kevin Boucher. |
