Complex hyperbolic space 
is a
complete Riemannian manifold with sectional curvature pinched between
-2 and -1/2.
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 is the Lie group PU(1,d).
G. D. Mostow [19] established that finite covolume discrete
subgroups of PU(1,d) 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 
for all 
.
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 PU(1,d) can be
classified as loxodromic, parabolic or elliptic. Let 
be a discrete group
generated by n loxodromic elements. The Dirichlet domain
(fundamental polyhedron) of 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
4+d(nd+2n-6).
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, §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 
, depending 
on t, i.e.\
with 
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.
In particular, the limit sets 
and 
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 1.1 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
are quasiconformally conjugate on 
.
The method used in the present paper yields analogous results for
Schottky groups in for 
.
Use the ball 
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 
, ) 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.
