## Parametrization of Möbius groups acting in a disk.(English)Zbl 0606.30042

We parametrize groups of Möbius-transformations by multipliers of the elements of the group. Let G be a group generated by 2g, $$g>1$$, hyperbolic Möbius-transformations mapping the upper half-plane U onto itself. Supposing that the set of generators of G satisfies one relation and certain technical conditions we construct a set of 6g-4 elements of G such that the multipliers of these Möbius-transformations parametrize G up to conjugation by a Möbius-transformation. The group G need not be discontinuous.
In the case of a Fuchsian group G we can apply these considerations to get a parametrization for the Teichmüller space of compact genus g Riemann surfaces by 6g-4 geodesic length functions.

### MSC:

 30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 14H15 Families, moduli of curves (analytic)

### Keywords:

Möbius groups; parametrization; geodesic length functions
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