##

Teichmüller spaces of Fuchsian groups

From now onwards, unless stated otherwise, we will assume that is a
finitely generated, torsion free Fuchsian group of the first kind,
leaving the upper half plane invariant (the theory we explain below
works in greater generality than this setting).

We define the space of bounded Beltrami coefficients for on the upper
half plane by

If is in the unit ball of
(we will
denote this unit ball by
), let
denote the solution of the Beltrami equation, fixing , 0 and
pointwise, obtained by extending to the lower half plane by
reflection. Let
be
the group homomorphism given by
.
Assume and are elements of
such
that the groups and are equal; set
,
. We have functions
induced by and respectively.

We define the **Teichmüller** and **Riemann ** spaces of as
the quotients of the unit ball
by the
relations and , defined as follows:

if
;

if
and are conjugate subgroups of
.

To define the modular group of we need to set up some notation:

We set
and
the **modular group** of by
. If
then the groups and are isomorphic
(to obtain that is isomorphic to , one has to
quotient by since any homeomorphism of has -lifts
to
).
The space has a pseudo-metric given by

where is a qc mapping, and and
coincide on
the real axis.

**Prop 2.12**
*The pseudo-metric on
is actually a metric. Moreover, if and in with
, then for any given such that
, there exists such that
and
.
is a complete metric on .*

**Prop 2.13**
*If is a (non-elementary, torsion free) Fuchsian group and
, then there exists a canonical isometry between
and .*

**Theorem 2.14**
*The modular group () acts properly
discontinuously by isometries on ( respectively).*
The proof of the theorem for is based on the next two results.
For a surface one uses the isometries between and the
identification between the groups and .

**Lemma 2.15**
*If is Fuchsian, non-cyclic, then the normalizer of
in
, , is also Fuchsian. Moreover, if
is of the firs kind then is a finite group, isomorphic to
the automorphism group of the surface
.*

**Lemma 2.16**
*If is Fuchsian of the first kind then the set of
traces squares of elements of is a discrete subset of
.*

Pablo Ares Gastesi
2005-08-31