Deformation of Schottky groups in complex hyperbolic space

Abstract

Let G = P U ( 1 , d ) G=PU(1,d) be the group of holomorphic isometries of complex hyperbolic space H C d \mathbf {H}^d_\mathbf {C} . The latter is a Kähler manifold with constant negative holomorphic sectional curvature. We call a finitely generated discrete group Γ = ⟨ g 1 , … , g n ⟩ ⊂ G \Gamma = \langle g_1,\dots , g_n \rangle \subset G a marked classical Schottky group of rank n n if there is a fundamental polyhedron for G G whose sides are equidistant hypersurfaces which are disjoint and not asymptotic, and for which g 1 , … , g n g_1, \dots , g_n are side-pairing transformations. We consider smooth families of such groups Γ t = ⟨ g 1 , t , … , g n , t ⟩ \Gamma _t = \langle g_{1,t}, \dots , g_{n,t} \rangle with g j , t g_{j,t} depending smoothly ( C 1 C^1 ) on t t whose fundamental polyhedra also vary smoothly. The groups Γ t \Gamma _t are all algebraically isomorphic to the free group in n n generators, i.e. there are canonical isomorphisms ϕ t : Γ 0 → Γ t \phi _t: \Gamma _0\to \Gamma _t . We shall construct a homeomorphism Ψ t \Psi _t of H ¯ C d = H C d ∪ ∂ H C d \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: ϕ t ( g ) ∘ Ψ t = Ψ t ∘ g ∀ g ∈ Γ 0 , 0 ≤ t ≤ 1 \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 ∂ H C d \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 D_0 onto D t D_t , where D t D_t is a fundamental domain of Γ t \Gamma _t . In two steps, this is extended equivariantly to H ¯ C d \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.