Global real analytic length parameters for Teichmüller spaces
The Teichmüller space, \(T(g)\), comprising conjugacy classes of marked Fuchsian groups of type \((g,0,0)\), is a bounded complex analytic manifold of complex dimension \(3g-3\), for each genus \(g \geq 2\). In this paper an explicit description of \(T(2)\) as a real analytic manifold is shown, utilizing seven ``length parameters'' (which are the absolute value of traces of certain chosen elements in the varying Fuchsian groups). The seven parameters (and it is known that six will not suffice for a global parametrization), are related by one algebraic relationship, and are subject to certain simple inequalities. The author claims that a similar result is possible, by induction, for a real analytic description of \(T(g)\), \(g\geq 3\) using \(6g-5\) length parameters. That is not shown in this article. One difficulty in reading the present paper is that some calculations rely on results quoted by the author from another manuscript [`Geometry of Möbius transformations acting on the hyperbolic plane and characterization of simple dividing loops on Riemann surfaces'] which is asserted to be ``in preparation''.
scientific article; zbMATH DE number 897340