Deformation of Schottky groups in complex hyperbolic space

Beat Aebischer - Robert R. Miner

Abstract:

Let be the group of holomorphic isometries of complex hyperbolic space . The latter is a Kähler manifold with constant negative holomorphic sectional curvature. We call a finitely generated discrete group a Schottky group of rank n if the sides of its Dirichlet domain (with respect to some point) are disjoint and not asymptotic. We consider smooth families of such groups ) on . The groups are all algebraically isomorphic to the free group in n generators, i.e. there are canonical isomorphisms . We shall construct a homeomorphism which is equivariant with respect to these groups:

which is quasiconformal on with respect to the Heisenberg metric, and which is symplectic in the interior. As a corollary, the limit sets of Schottky groups of equal rank are quasiconformally equivalent to each other.
The main tool for the construction is a time-dependent Hamiltonian vector field used to define a diffeomorphism, mapping onto . In two steps, this is extended equivariantly to .

The method yields similar results for real hyperbolic space, while the analog for the other rank-one symmetric spaces of noncompact type cannot hold.

Robert Miner
Sat May 24 18:07:32 CDT 1997