| In these two talks we shall give two different self-contained perspectives on the following problem: Let M be a closed negatively curved manifold with fundamental group G. The Cartan-Hadamard theorem provides a natural boundary sphere S for the universal cover of M. Let H be a quasiconvex subgroup of G (e.g. the fundamental group of a totally geodesic submanifold). Then H gives rise to a G-invariant collection P of closed subsets of S. P is called a (symmetric) pattern. If a quasiconformal homeomorphism of S preserves P as a set, is it conformal? More generally what is the group of homeomorphisms of S preserving P?
The first talk will give a positive answer to this problem when M has constant negative curvature and H is any quasiconvex subgroup of infinite index in G.
The second talk will give a positive answer to the problem when M is an arbitrary negatively curved manifold (or more generally a Poincare duality Gromov-hyperbolic group) and H is special in a sense that will be explained. The problem originates in Mostow rigidty. |