Semialgebraic Description of Teichmiiller Space

Abstract

We give a concrete semialgebraic description of Teichmiiller space $T_g$ of the closed surface group ${\Gamma }_g$ of genus $g(\ge 2)$. Our result implies that for any ${\text{SL}}_2(\mathbf{R})$-representation of ${\Gamma }_g$, we can determine whether this representation is discrete and faithful or not by using $4g-6$ explicit trace inequalities. We also show the connectivity and contractibility of $T_g$ from the point of view of ${\text{SL}}_2(\mathbf{R})$-representations of ${\Gamma }_g$. Previously, these properties of _Tg _had been proved by using hyperbolic geometry and quasi-conformal deformations of Fuchsian groups. Our method is simple and only uses topological properties of the space of ${\text{SL}}_2(\mathbf{R})$-representations of ${\Gamma }_g$.