An isomorphism theorem for Teichmuller curves

It is proved that for any Fuchsian group Gamma such that H/Gamma is a hyperbolic Riemann surface, the Teichmuller curve V(Gamma) has a unique complex manifold structure so that the natural projection of the Bers fiber space F(Gamma) onto V(Gamma) is holomorphic with local holomorphic sections. An isomorphism theorem for Teichmbller curves is deduced, which generalizes a classical result that the Teichmuller curve V(Gamma) depends only on the type of Gamma and not on the orders of the elliptic elements of Gamma when H/Gamma is a compact hyperbolic Riemann surface.