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Homomorphisms of Fuchsian groups to PSL(2,\({\mathbb{R}})\). (English) Zbl 0598.57007

Let \(\Gamma =\Gamma (g;\alpha_ 1,...,\alpha_ n)\) be a cocompact Fuchsian group of genus g with branch indices \(\alpha_ 1,...,\alpha_ n\). In the present paper the set of components of Hom(\(\Gamma\),PSL(2, \({\mathbb{R}}))\) is determined. Let e: Hom(\(\Gamma\),PSL(2, \({\mathbb{R}}))\to H^ 2(\Gamma; {\mathbb{Z}})\) be defined as follows: \(e(f)=f^*(c)\), where \(f^*: H^ 2(PSL(2, {\mathbb{R}}); {\mathbb{Z}})\to H^ 2(\Gamma; {\mathbb{Z}})\) is induced by f and c is a generator of \(H^ 2(PSL(2, {\mathbb{R}}); {\mathbb{Z}})\cong {\mathbb{Z}}\). Then it is shown that the fibers of the map e are the components of Hom(\(\Gamma\),PSL(2, \({\mathbb{R}}))\), and the image of e in \(H^ 2(\Gamma; {\mathbb{Z}})\) is exactly determined (the case \(g>0\) has been known previously). Also, the connection with the question which Seifert circle bundles over a surface of genus g admit a transverse foliation is discussed. By an example it is shown that the case \(g=0\) is more subtle than expected from the known solution in the case \(g>0\).
Reviewer: B.Zimmermann

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57R30 Foliations in differential topology; geometric theory
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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