## 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.

Prop 2.11   The following are equivalent:
(i) is homotopic to ;
(ii) ;
(iii) , for all .

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 .

Proof. By conjugation we can assume that is normalized; that is, , 0 and are points in the limit set of . Let be an element of ; then we assigned to it (the Teichmüller class of) the marked Riemann surface , where is induced by as above.

For the inverse mapping, given a marked surface , write ; then lifts to such that , and fixes , 0 and pointwise. We assign to the given marked Riemann surface the Beltrami coefficient .

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