Teichmüller space and fundamental domains of Fuchsian groups (Q1594947)

A closed Riemann surface of genus \(g\geq 2\) has a canonical fundamental domain which is a polygon with \(4g\) sides in the upper half plane. It is known that the Teichmüller space \(T_g\) of closed Riemann surface of genus \(g\) is constructed by these polygons. In this paper the author considers a new polygon \(P(g)\) with \(4g\) sides and constructs \(T_g\). The property of the polygon \(P(g)\) is that the opposite sides have the same length and are identified in the Riemann surface. It is used to give a new proof that \(T_g\) is homeomorphic to \({\mathbb R}^{6g-6}\). As an application of the polygon \(P(g)\) the author shows that \(T_g\) for \(g=2\) can be parameterized by \(7(=6g-5)\) geodesic length functions, taken as homogeneous parameters. Also the author gives necessary and sufficient conditions for a closed hyperbolic surface \(M\) of genus \(g\) to be hyperelliptic, for example a condition that the polygon \(P(g)\) corresponding to \(M\) has equal opposite angles.