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

Seppälä, Mika; Sorvali, Tuomas

Comment. Math. Helv. 61, 149-160 (1986).

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