Beat Aebischer
-
Robert R. Miner
Let G=PU(1,d) 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 
with 
depending smoothly ( 
) 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 
of 
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
, where is the Dirichlet domain of .
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.