Jankins, Mark; Neumann, Walter Homomorphisms of Fuchsian groups to PSL(2,\({\mathbb{R}})\). (English) Zbl 0598.57007 Comment. Math. Helv. 60, 480-495 (1985). 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 Cited in 11 Documents 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) Keywords:Seifert fibration; Seifert fibre space; PSL(2,\({\mathbb{R}})\); cocompact Fuchsian group; Seifert circle bundles; transverse foliation PDFBibTeX XMLCite \textit{M. Jankins} and \textit{W. Neumann}, Comment. Math. Helv. 60, 480--495 (1985; Zbl 0598.57007) Full Text: DOI EuDML