Semialgebraic description of Teichmüller space

We give a concrete semialgebraic description of Teichmüller space T_g of the closed surface group Γ_g of genus g(≥2). Our result implies that for any SL₂(R)-representation of Γ_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 SL₂(R)-representations of Γ_g. Previously, these properties of T_g 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 SL₂(R)-representations of Γ_g.