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Conformal Geometry and Dynamics

Published by the American Mathematical Society since 1997, the purpose of this electronic-only journal is to provide a forum for mathematical work in related fields broadly described as conformal geometry and dynamics. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.49.

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Deformation of Schottky groups in complex hyperbolic space
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by Beat Aebischer and Robert Miner
Conform. Geom. Dyn. 3 (1999), 24-36
Published electronically: March 11, 1999


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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Bibliographic Information
  • Beat Aebischer
  • Affiliation: Leica AG, PPT 4199, 9435 Heerbrugg, Switzerland
  • Email:
  • Robert Miner
  • Affiliation: The Geometry Center, University of Minnesota, Minneapolis, Minnesota 55454
  • Email:
  • 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
  • © Copyright 1999 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 3 (1999), 24-36
  • MSC (1991): Primary 30C65; Secondary 32G10, 57S30, 53C55, 58F05
  • DOI:
  • MathSciNet review: 1677557