Poincaré series and teichmüller space

Abstract

We consider the Poincare series operator Θ as a holomorphic mapping of the Teichmuller space T(Γ) of a Fuchsian group Γ into the Bers space A(Γ) of integrable holomorphic quadratic forms, and show that if Γ is finitely generated and of the first kind, then ⊖ gives a local coordinate of T(Γ) at the origin. Conversely, if Θ(T(Γ)) contains an interior point in A(Γ), then Γ is finitely generated and of the first kind.