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
denote this unit ball by
denote the solution of the Beltrami equation, fixing , 0 and
pointwise, obtained by extending to the lower half plane by
the group homomorphism given by
Assume and are elements of
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
relations and , defined as follows:
and are conjugate subgroups of
To define the modular group of we need to set up some notation:
the modular group of by
then the groups and are isomorphic
(to obtain that is isomorphic to , one has to
quotient by since any homeomorphism of has -lifts
The space has a pseudo-metric given by
where is a qc mapping, and and
the real axis.
The pseudo-metric on
is actually a metric. Moreover, if and in with
, then for any given such that
, there exists such that
is a complete metric on .
If is a (non-elementary, torsion free) Fuchsian group and
, then there exists a canonical isometry between
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 .
If is Fuchsian, non-cyclic, then the normalizer of
, , is also Fuchsian. Moreover, if
is of the firs kind then is a finite group, isomorphic to
the automorphism group of the surface
If is Fuchsian of the first kind then the set of
traces squares of elements of is a discrete subset of
Pablo Ares Gastesi