Curvature, uniformly bounded representations, and classifying hyperbolic groups
by
Michael Cowling
University of New South Wales

The aim of this talk is to illustrate a conjectured possible classification of some hyperbolic groups, based on uniformly bounded representations, and its possible connections to geometrical invariants.

It is known that hyperbolic groups \Gamma may be divided into two classes: those with and those without Kazhdan's Property T. By the Delorme-Guichardet Theorem, those which have Property T do not have unitary representations with cohomology. Yehuda Shalom conjectures that these groups do have (a small number of) uniformly bounded representations with cohomology. In this case, it will be possible to use the minimal norm of the (worst) representation with cohomology to associate a number with the group \Gamma. This will presumably to linked to the invariant \Lambda(\Gamma) produced by Haagerup and his co-workers. By looking at the case of lattices in Sp(n, 1), it is also possible to conjecture that the proposed invariant is related to the curvature of the hyperbolic group (with the word metric).

Date received: November 26, 2001