Isomorphisms Of Fiber Spaces Over Teichmüller Spaces

Let Г be a Fuchsian group acting on the upper half plane H, and H_Г be H with all of the fixed points of elliptic elements of Г removed. We will be mainly concerned with the isomorphisms between some important fiber spaces over the Teichmuller space T(Г), including the Bers fiber space F(Г), the "punctured" fiber space F₀(Г), the Teichmuller curve V(Г) and the "punctured" Teichmuller curve V₀(Г).We first prove that a conformal mapping from H_Г1/Г₁ onto H_Г2/Г₂ induces a biholo-morphic isomorphism from V₀(Г₁) onto V₀(Г₂), which implies that the structure of V₀(Г) depends only on the topological type but not on the signature of Г when Г is finitely generated and of the first kind.We will also determine the biholomorphic isomorphisms between Bers ( or "punctured" ) fiber spaces and those between ( "punctured" ) Teichmuller curves. We first show that a biholomorphic isomorphism between Bers fiber spaces ( or Teichmuller curves ) is an allowable mapping when a Fuchsian group Г contains no elliptic elements and parabolic elements ; then we show that a fiber-preserving biholomorphic isomorphism between Bers fiber spaces ( or Teichmuller curves ) is an allowable mapping while Г contains no elliptic elements and is not of type (0,3) and (1,1); finally we show that a fiber-preserving biholomorphic isomorphism between "punctured" fiber spaces is an allowable mapping for any Fuchsian group Г not of type (0,3) and (1,1). In particular, a biholomorphic automorphism is always induced by an element of the extended modular group or of the modular group in all cases.