International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 4, Pages 695-708
doi:10.1155/S0161171293000870

On monodromy map

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.