Complex Hyperbolic triangle groups
by
Rich Schwartz
Univ of Maryland

The usual triangle groups are generated by reflections in the sides of a hyperbolic triangle. I will talk about the problem of deforming these groups into complex hyperbolic space. I'll try to cover three things:

1. An exact characterization of which deformations of the ideal triangle group are discrete and faithful. (This is a solution to the so called Goldman-Parker conjecture.)

2. The proof that the manifold at infinity, for the "last discrete" complex hyperbolic ideal triangle group, is commensurable to the Whitehead link complement.

3. Some examples of deformations of triangle groups where the manifold at infinity is homeomorphic to a closed hyperbolic 3-manifold. (As far as I know these give the first examples of "complete spherical CR structures" on hyperbolic 3-manifolds.)

This work was inspired by computer experimentation. I'll try to bring in some nice computer pictures, and also have available some of the software I wrote for exploring and visualizing these groups. I'll also talk about some computer inspired conjectures about the limit sets of these groups, the poincare series, the length spectrum, etc.

Date received: March 10, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caca-65.