Jharna D. Sengupta

Copyright © 1993 Jharna D. Sengupta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let Γ be a Fuchsian group acting on the upper half-plane U and having signature {p,n,0;v1,v2,…,vn}; 2p−2+∑j=1n(1−1vj)>0.

Let T(Γ) be the Teichmüller space of Γ. Then there exists a vector bundle ℬ(T(Γ)) of rank 3p−3+n over T(Γ) whose fibre over a point t∈T(Γ) representing Γt is the space of bounded quratic differentials B2(Γt) for Γt. Let Hom(Γ,G) be the set of all homomorphisms from Γ into the Mbius group G.

For a given (t,ϕ)∈ℬ(T(Γ)) we get an equivalence class of projective structures and a conjugacy class of a homomorphism x∈Hom(Γ,G). Therefore there is a well defined map Φ:ℬ(T(Γ))→Hom(Γ,G)/G, Φ is called the monodromy map. We prove that the monromy map is hommorphism. The case n=0 gives the previously known result by Earle, Hejhal Hubbard.