Teichmüller space of genus two based on Schmutz Schaller's hyperbolic polygons

Abstract

Following the idea of P. Schmutz Schaller, we shall consider a parametrization of the Teichmüller space $\mathscr{T}_2$ of compact Riemann surfaces of genus two. In the first part of this paper, we calculate the coordinates of 4 kinds of surface uniformized by Fuchsian groups whose fundamental regions can be the regular octagon. In the second part, we give a characterization of $\mathscr{T}_2$ in R⁷.