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International Journal of Mathematics and Mathematical Sciences
Volume 16, Issue 4, Pages 695-708

On monodromy map

Department of Mathematics and Computer Science, Elizabeth City State University, Elizabeth City, NC 27909, USA

Received 2 April 1992; Revised 22 September 1992

Copyright © 1993 Hindawi Publishing Corporation. 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.


Let Γ be a Fuchsian group acting on the upper half-plane U and having signature {p,n,0;v1,v2,,vn}; 2p2+j=1n(11vj)>0.

Let T(Γ) be the Teichmüller space of Γ. Then there exists a vector bundle (T(Γ)) of rank 3p3+n over T(Γ) whose fibre over a point tT(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)(T(Γ)) we get an equivalence class of projective structures and a conjugacy class of a homomorphism xHom(Γ,G). Therefore there is a well defined map Φ:(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.