Deformation of Schottky groups in complex hyperbolic space

Abstract:

Let $G=PU(1,d)$ be the group of holomorphic isometries of complex hyperbolic space $\mathbf {H}^d_\mathbf {C}$. The latter is a Kähler manifold with constant negative holomorphic sectional curvature. We call a finitely generated discrete group $\Gamma = \langle g_1,\dots , g_n \rangle \subset G$ a marked classical Schottky group of rank $n$ if there is a fundamental polyhedron for $G$ whose sides are equidistant hypersurfaces which are disjoint and not asymptotic, and for which $g_1, \dots , g_n$ are side-pairing transformations. We consider smooth families of such groups $\Gamma _t = \langle g_{1,t}, \dots , g_{n,t} \rangle$ with $g_{j,t}$ depending smoothly ($C^1$) on $t$ whose fundamental polyhedra also vary smoothly. The groups $\Gamma _t$ are all algebraically isomorphic to the free group in $n$ generators, i.e. there are canonical isomorphisms $\phi _t: \Gamma _0\to \Gamma _t$. We shall construct a homeomorphism $\Psi _t$ of $\overline {\mathbf {H}}^d_\mathbf {C} = \mathbf {H}^d_\mathbf {C}\cup \partial \mathbf {H}^d_\mathbf {C}$ which is equivariant with respect to these groups: \begin{equation*} \phi _t(g) \circ \Psi _t = \Psi _t \circ g \quad \; \forall g\in \Gamma _0, \quad 0\leq t\leq 1 \end{equation*} which is quasiconformal on $\partial \mathbf {H}^d_\mathbf {C}$ with respect to the Heisenberg metric, and which is symplectic in the interior. As a corollary, the limit sets of such 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 $D_0$ onto $D_t$, where $D_t$ is a fundamental domain of $\Gamma _t$. In two steps, this is extended equivariantly to $\overline {\mathbf {H}}^d_\mathbf {C}$. The method yields similar results for real hyperbolic space, while the analog for the other rank-one symmetric spaces of noncompact type cannot hold.