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Deformation of Schottky Groups in Complex Hyperbolic Space

Beat Aebischer

Leica AG, PPT 4199, 9435 Heerbrugg, Switzerland

Robert Miner

The Geometry Center, University of Minnesota, Minneapolis, MN 55454

Feb 15, 1997.

Abstract

Let G=PU(1,d) be the group of holomorphic isometries of complex hyperbolic space HCd. The latter is a Kähler manifold with constant negative holomorphic sectional curvature. We call a finitely generated discrete group Γ=⟨ g1 ,..., gn ⟩⊂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 g1 ,..., gn are side-pairing transformations. We consider smooth families of such groups Γt =⟨ g1,t ,..., gn,t ⟩ with gj,t depending smoothly ( C1 ) on t whose fundamental polyhedra also vary smoothly. The groups Γt are all algebraically isomorphic to the free group in n generators, i.e. there are canonical isomorphisms φt : Γ0 → Γt . We shall construct a homeomorphism Ψt of H&OverLine; Cd = HCd ∪∂ HCd which is equivariant with respect to these groups:
φt (g)∘ Ψt = Ψt ∘g&medsp;&medsp;&medsp;&medsp;&medsp;∀g∈ Γ0 ,&medsp;&medsp;&medsp;0≤t≤1

which is quasiconformal on ∂ HCd 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 D0 onto Dt , where Dt is a fundamental domain of Γt . In two steps, this is extended equivariantly to H&OverLine; Cd .

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