Semialgebraic Description of Teichmiiller Space

Abstract

We give a concrete semialgebraic description of Teichmiiller space Tg of the closed surface group Γ_g of genus g(≥ 2). Our result implies that for any SL_2(R)-representation of Γ_g, we can determine whether this representation is discrete and faithful or not by using 4_g—6 explicit trace inequalities. We also show the connectivity and contractibility of Tg from the point of view of SL_2(R)-representations of Γ_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 SL_2(R)-representations of Γ_g.__