Complex hyperbolic space is a complete Riemannian manifold with sectional curvature pinched between -2 and . A standard model for it is the unit ball ) with the Bergman metric. For the case d=1 note that is isomorphic to the real hyperbolic plane . As a complex manifold, admits a natural Kähler structure, and therefore also a symplectic structure.
The group of holomorphic isometries of [tex2html_wrap_inline673] is the Lie group [tex2html_wrap_inline675]. G. D. Mostow [19] established that finite covolume discrete subgroups of are rigid. Recent work has shown that interesting phenomona still arise in connection with deformations of smaller discrete subgroups. (See for example [8], [9].)
At the same time, a successful theory of quasiconformal mapping on complex hyperbolic space and its boundary has been developed by Korányi, Reimann, Pansu and others. (See for example [15] and [20].) Moreover, as in real hyperbolic geometry, there seems to be much fertile ground for the application of quasiconformal mappings to deformation questions about complex hyperbolic groups. Indeed, Mostow's rigidity theorem involves quasiconformal mapping in a vital way. In this article, we offer another application of quasiconformal mapping to deformation theory of groups that are far from having finite covolume. Specifically, we show that smooth deformations of complex hyperbolic Schottky groups are quasiconformally stable in an appropriate sense.
For comparison recall the definition of quasiconformal stability from the theory of Kleinian groups as given by Bers in [2]. Let G be any subgroup of . A homomorphism will be called allowable if it preserves the square traces of parabolic and elliptic elements. Say that is a quasiconformal deformation of G if there exists a quasiconformal mapping such that . A finitely generated Kleinian group is then called quasiconformally stable if every allowable homomorphism sufficiently close to the identity is a quasiconformal deformation. In [2], Bers described a criterion (involving the quadratic differentials for ) to determine whether a group is quasiconformally stable. As a consequence of this criterion, it follows that Fuchsian groups, Schottky groups, groups of Schottky type and certain non-degenerate B-groups are all quasiconformally stable [3].
As with real hyperbolic isometries, elements of can be classified as loxodromic, parabolic or elliptic. Let be a discrete group generated by n loxodromic elements. The Dirichlet domain (fundamental polyhedron) of [tex2html_wrap_inline709] with respect to the point is given by
In analogy with Kleinian groups, if the sides of D do not intersect and are not asymptotic, we call a Schottky group of rank n. The space of such groups (modulo conjugation) has real dimension . Namely, by [4, Lemma 3.2.2,], a loxodromic element with given fixed points has real parameters left. Every fixed point contributes 2d-1 real parameters, and three fixed points can be normalized by Cartan's angular invariant (see e.g. [7, ,]). The limit set
of such a group is closed and totally disconnected (i.e. a Cantor set) (see e.g. [21, Theorem 12.1.21,]).
We consider a family of Schottky groups with generators on t, i.e.\ of class . Note that the are all isomorphic to the free group on n generators (cf. [21, Theorem 12.1.19,]), hence there are canonical isomorphisms
We show that smooth deformations of Schottky groups are quasiconformally stable in the following sense.
Note that the boundary of complex hyperbolic space can be identified with the one-point compactification of the Heisenberg group , see e.g. [7, § 2.1,], [13] or the next section. The Heisenberg metric on is also defined in the next section.
Because two Schottky groups of the same rank can clearly be connected by a smooth family of such groups, we have the following corollary.
[cor123]
In particular, the limit sets
are
quasiconformally equivalent:
.
Because the image of a Schottky group under a homomorphism close to
the identity is again a Schottky group of the same rank, the
quasiconformal stability expressed by Theorem 
includes
quasiconformal stability in the sense of Bers.
In real hyperbolic space
, the
following is known.
For d=3, members of a continuously parametrized family of
finitely generated discrete groups are canonically isomorphic to
each other, [12, Theorem 3,].
Again for d=3, using the generalized
-lemma of holomorphic
motions, one can show:
If
is a holomorphic family of finitely
generated Kleinian groups such that no new parabolic elements are
created as z varies in the unit disk, then the
are
canonically quasiconformally conjugate on their limit sets,
[1].
In fact, the equivariant version of the generalized
-lemma
by Earle, Kra and Krushkal' [6] shows that the groups
.
The method used in the present paper yields analogous results for
Schottky groups in
.
Use the ball [tex2html_wrap_inline807] as a model for real hyperbolic
space and on its boundary
use the euclidean (chordal)
metric to define quasiconformality.
In the interior, it does not matter whether one uses the hyperbolic
or the euclidean metric to define quasiconformality, because these
metrics are conformally equivalent.
As a corollary, Schottky groups of equal rank n (acting on [tex2html_wrap_inline831], [tex2html_wrap_inline833]) are quasiconformally
conjugate. The hyperbolic quotient manifolds
and
are quasiconformally equivalent.
By [17, VIII.D.1,], for d=3 these manifolds are `handlebodies
of genus n'.
By the known rigidity of quaternionic hyperbolic space and the
Cayley plane, analogous results for Schottky groups on these
spaces cannot hold. Namely, as P. Pansu has shown, every
quasiconformal map on the boundary of quaternionic hyperbolic space or
the Cayley plane is the extension of an isometry (Corollary 11.2 and
Proposition 11.5 in [20]). Thus, for the result to hold, the
space of Schottky groups modulo conjugation by isometries would have
to reduce to a point, which of course is not the case.
However, in the quaternionic and Cayley cases our construction still
yields an equivariant homeomorphism of hyperbolic space and its boundary
which is a diffeomorphism in the interior.
In the general case of negatively curved groups, Martin [16]
has shown the existence of a quasisymmetric conjugacy on the limit
set.
Section 2 contains introductory material and some lemmas which are
used in section 3 to proof the theorems.
It is a pleasure for us to thank Martin Reimann for many helpful
discussions.
Next: Notation and preliminaries
Up: Deformation of Schottky groups
Previous: Deformation of Schottky groups
Robert Miner
Sat May 24 18:07:32 CDT 1997