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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173

 

Deformation of Schottky groups
in complex hyperbolic space


Authors: Beat Aebischer and Robert Miner
Journal: Conform. Geom. Dyn. 3 (1999), 24-36
MSC (1991): Primary 30C65; Secondary 32G10, 57S30, 53C55, 58F05
Published electronically: March 11, 1999
MathSciNet review: 1677557
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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.


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Additional Information

Beat Aebischer
Affiliation: Leica AG, PPT 4199, 9435 Heerbrugg, Switzerland
Email: Beat.Aebischer@email.leica.com

Robert Miner
Affiliation: The Geometry Center, University of Minnesota, Minneapolis, Minnesota 55454
Email: rminer@geom.umn.edu

DOI: http://dx.doi.org/10.1090/S1088-4173-99-00010-7
PII: S 1088-4173(99)00010-7
Keywords: Complex hyperbolic space, Schottky group, deformation, quasiconformal mapping, Heisenberg group
Received by editor(s): March 3, 1997
Received by editor(s) in revised form: November 4, 1998
Published electronically: March 11, 1999
Additional Notes: B. Aebischer supported by Schweizerischer Nationalfonds
R. Miner partially supported by NSF grant DMS-9404174
Article copyright: © Copyright 1999 American Mathematical Society