Parametrization of Möbius groups acting in a disk (Q1084551)

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.