Deformation of Schottky groups in complex hyperbolic space

Abstract

Abstract. Let G = P U(1, d) be the group of holomorphic isometries of complex hyperbolic space H d C . The latter is a Kähler manifold with constant negative holomorphic sectional curvature. We call a finitely generated discrete group Γ = g 1 , . . . , 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 g 1 , . . . , gn are side-pairing transformations. We consider smooth families of such groups Γt = g 1,t , . . . , gn,t with g j,t depending smoothly (C 1 ) 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 : Γ 0 → Γt. We shall construct a homeomorphism Ψt ofwhich is equivariant with respect to these groups:which is quasiconformal on ∂H d 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 Dt, where Dt is a fundamental domain of Γt. In two steps, this is extended equivariantly toThe method yields similar results for real hyperbolic space, while the analog for the other rank-one symmetric spaces of noncompact type cannot hold.

The geometry of complex hyperbolic packs

“…Complex hyperbolic plane H 2 C is a 2-complex dimensional complete Kähler manifold with constant holomorphic sectional curvature −1 and real sectional curvature pinched between −1/4 and −1. Its group of holomorphic isometries is PU (2,1). Real hyperbolic plane is embedded into H 2 C in two ways.…”

“…Teichmüller space T ( ) comprises of Fuchsian representations of π 1 , that is discrete, faithful, type preserving and geometrically finite representations of π 1 into SU (1,1). Every such representation is then homotopically equivalent to a quasiconformal representation; in this case, by uniformisation we may identify π 1 with a Fuchsian group and then any ρ ∈ T ( ) is equivalent to one of the form…”

“…This theory is the analogous to our case, of the Ahlfors-Bers theory of quasiconformal mappings of the complex plane and we have to use it in order to associate a quasiconformal deformation of the Heisenberg group to a deformation of a fundamental polyhedron. To that direction little progress has been made so far; for instance, Aebischer and Miner proved this result for the elementary case of complex hyperbolic quasi-Fuchsian space of a classical complex hyperbolic Schottky group of n generators, see [1]. Such a group admits a fundamental domain whose sides are disjoint bisectors.…”

“…There exist only 2-dimensional totally geodesic submanifolds of H 2 C : (a) Complex lines L which have constant curvature −1. These submanifolds realise isometric embeddings of H 1 C into H 2 C . Every complex line L is the image under some A ∈ SU(2, 1) of the complex line…”

“…Denote by J the natural complex structure of H 2 C . A symplectomorphism F defines another complex structure J μ in H 2 C by the relation [4, lemma 7.5], there is a complex antilinear self mapping of the holomorphic tangent bundle T (1,0) of the complex hyperbolic plane such that the holomorphic tangent bundle T (1,0) μ of the J μ complex structure is T (1,0) where…”