Extended monodromy for bordered surfaces
Let G be a finitely generated Fuchsian group acting without fixed points on the upper half-plane U, \(B_ 2(G)\) the vector bundle of bounded quadratic differentials over the Teichmüller space T(G) and \({\mathcal M}: B_ 2(G)\to Hom'(G),Aut {\hat {\mathbb{C}}})\) the monodromy mapping. For bordered surfaces \(S=U/G\) one can restrict \({\mathcal M}\) to the space \(B^{\#}_ 2(G)\) of bounded quadratic differentials which are real on the boundary. A previous example of the authors [Kleinian groups and related topics, Proc. Workshop, Oaxtepec/Mex. 1981, Lect. Notes Math. 971, 36-47 (1983; Zbl 0527.30030)] proves that (i) there may be non-manifold-points in the image of \({\mathcal M}\) and (ii) \({\mathcal M}\) is not one-to-one on every fiber. Now it is shown that (iii) \({\mathcal M}\) is not locally one-to-one. It is still open, whether it is locally one-to-one on fibers or not. Let \(S^ d=U/G^ d\) be the Schottky double of S and \(G^{d+}\) an index two extension of \(G^ d\) in the full group \(Aut^+{\hat {\mathbb{C}}}\) of conformal and anticonformal automorphisms of \({\hat {\mathbb{C}}}\) such that S is conformally equivalent to the interior of \(U/G^{d+}\). Then the extended monodromy mapping \({\mathcal M}_{ext}: B^{\#}_ 2(G)\to Hom'(G^{d+},Aut^+ {\hat {\mathbb{C}}})\) is naturally defined. The following is shown to hold: (i) \({\mathcal M}_{ext}\) is a real-analytic mapping with values in a manifold of the same dimension as \(B^{\#}_ 2(G);\) (ii) \({\mathcal M}_{ext}\) is injective on every fiber of \(B^{\#}_ 2(G);\) (iii) \({\mathcal M}_{ext}\) is nonsingular and (iv) \({\mathcal M}_{ext}\) is not globally one-to-one.
scientific article; zbMATH DE number 4175354
scientific article