Proof of Theorem 1.1.
We have Schottky groups 
.
Set 
; then the sides of the Dirichlet
domain 
of 
are
By lemma 2.1 we have time dependent Hamiltonians
generating isometries 
mapping 
onto
for 
. If we set
then 
is an isometry mapping 
onto 
for
.
Since is symplectic, there exists a global time dependent
Hamiltonian 
for it as in the proof of
lemma 2.1.
For 
choose cutoff functions
which are 
in the
first and
in the second argument. Moreover, require that
vanishes outside some euclidean neighborhood of , that it is
identically one in a smaller neighborhood, and that the supports of
the 
are disjoint.
(Remember the sides of do not intersect, where is the
Dirichlet domain for centered at 
.)
We can also assume that
for .
In fact, it suffices to choose smooth functions 
with sufficiently small supports and then define
by these conditions.
The diffeomorphism 
generated by the
time dependent Hamiltonian
then maps each side of 
onto the corresponding side of .
Applying Lemma 2.2 to every summand in the above equation
(frozen to a fixed time t), we see that 
and hence
extends smoothly to the boundary 
.
Let
be a fundamental set for the action of on 
.
Adding the points on the boundary at infinity adhering to 
, we
get a fundamental set 
for the action of on
, where 
is the limit set of
. It follows that 
.
We can now extend 
to an equivariant map 
on 
as follows.
For 
there is a unique element h
of 
such that 
, we set
Away from the boundary of translates of the fundamental set,
is clearly a diffeomorphism. Further, on the neighborhood
of one has
Thus, the extension is indeed a diffeomorphism
.
Moreover, agrees with on a neighborhood of
, and hence
holds. One also easily verifies that the equivariance condition (1.1) is satisfied.
We now show extends to a homeomorphism of 
.
For Schottky groups a point in the limit set corresponds to a unique
infinite word in the generators (written from left to right).
That is, to the point 
corresponds the sequence
such that 
and 
is a word of length j
in the generators of .
If we set
it is clear that 
is one-to-one and continuous,
because nearby limit points correspond to words that have long equal
beginnings.
An arbitrary sequence in converging to
can be written (uniquely) as 
, with
and 
.
Then the line
shows that the extension is continuous on .
Changing the roles of the times 0 and t yields the same for
.
Thus is a homeomorphism.
Finally, it remains to show that is quasiconformal with
respect to the Heisenberg metric on 
.
We begin by showing that is quasiconformal on
.
By Lemma 2.2, 
is
the [0,1]-flow of the continuously time dependent vector field
, where 
is
in its second argument and is given by
Proposition 25 in [15] (which holds as well for time dependent
potentials p) now implies
is K-quasiconformal with K depending only on the first and second
horizontal derivatives of the cutoff functions and the
first horizontal derivatives of 
. (Note that the second
horizontal derivatives of vanish, because the flow generated by
is conformal.) By the definition (3.1) of the
extension it immediately follows that is
quasiconformal on , since elements of
PU(1,d) operate conformally on .
To show that quasiconformality extends across 
, we use the
analytic definition [15]. Since has Lebesgue
measure zero, we only have to show the extension of to
has the ACL property, i.e. for every smooth fibration
by horizontal curves, is absolutely continuous on
almost every curve in .
Consider first the case where the limit set has zero
-dimensional Hausdorff measure. Then the set of curves
in meeting has measure zero. To see this,
note that by definition, the measure on has the property
see [15].
Hence 
inherits the ACL
property from
.
In the general case, we extend the family up to t=2 in such
a way that the generators of 
all preserve 
. The limit set 
then is contained
in 
and has Hausdorff dimension at most two (the direction
transverse to the contact hyperplane counts twice,
see [5, p. 526,] or [18]). In particular,
has zero -dimensional Hausdorff measure. Now set
. As in the special
case above, we can construct homeomorphisms 
with the required regularity properties and
such that
where 
.
One immediately verifies that 
satisfies (1.1) as well as the
regularity properties claimed in Theorem 1.1.
Proof of Theorem 1.3. The proof goes along the same lines as the previous one, except that one gets quasiconformality much easier. So we just sketch the differences.
The analog of Lemma 2.1 holds, where the vector
field 
is not Hamiltonian, but is defined on the closure
.
Instead of cutting off Hamiltonians we cut off the time dependent
vector fields 
, which generate isometries mapping
onto . Then the diffeomorphism
generated by the vector field 
maps each side
of the Dirichlet domain onto the corresponding side of .
The restriction is extended as before to an
equivariant diffeomorphism 
.
As before, extends as a homeomorphism to .
Now, is a diffeomorphism, hence is quasiconformal on
and on .
Hence the extension of is
quasiconformal on and on .
As at the end of the previous proof, we only need to consider the
case when is `small', for instance when it has
-finite (d-2)-dimensional Hausdorff measure.
Then Theorem 35.1 in [22] shows that is
quasiconformal on all of .
Alternatively, one can directly apply Theorem 4.2 of [11]
without having to make small.
