On monodromy map (Q689012)

Let \(\Gamma\) be a Fuchsian group acting on the upper half-plane \(U\) and having signature \(\{p,n,0;\nu_ 1,\nu_ 2,\) \(\ldots,\nu_ n\}\); \(2p- 2+\sum^ n_{j=1}\left(1-{1\over\nu_ j}\right)>0\). Let \(T(\Gamma)\) be the Teichmüller space of \(\Gamma\). Then there exists a vector bundle \({\mathcal B}(T(\Gamma))\) of rank \(3p-3+n\) over \(T(\Gamma)\) whose fibre over a point \(t\in T(\Gamma)\) representing \(\Gamma_ t\) is the space of bounded quadratic differentials \(B_ 2(\Gamma_ t)\) for \(\Gamma_ t\). Let \(\Hom(\Gamma,G)\) be the set of all homomorphisms from \(\Gamma\) into the Moebius group \(G\). For a given \((t,\varphi)\in{\mathcal B}(T(\Gamma))\) we get an equivalence class of projective structures and a conjugacy class of a homomorphism \(\chi\in\Hom(\Gamma,G)\). Therefore there is a well defined map \[ \Phi:{\mathcal B}(T(\Gamma))\to\Hom(\Gamma,G)/G, \] \(\Phi\) is called the monodromy map. We prove that the monodromy map is a holomorphic local homeomorphism. The case \(n=0\) gives the previously known result by Earle, Hejhal and Hubbard.