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Lifting surface groups to SL(2,\({\mathbb{C}})\). (English) Zbl 0531.30037
Kleinian groups and related topics, Proc. Workshop, Oaxtepec/Mex. 1981, Lect. Notes Math. 971, 1-5 (1983).
[For the entire collection see Zbl 0489.00012.]
The fundamental group of a compact orientable surface of genus \(g\geq 2\) is a subgroup of \(PSL(2,{\mathbb{R}})=SL(2,{\mathbb{R}})/\{\pm I\}\) with 2g generators and one relation: \(\prod_{i odd}[\gamma_ i,\gamma_{i+1}]=id.\) If \({\hat \phi}{}_ i\) is either of the two lifts of \(\gamma_ i\) to SL(2,\({\mathbb{R}})\), then \(\prod_{i odd}[{\hat \gamma}_ i,{\hat \gamma}_{i+1}]=\pm I.\) The authors prove that for any choice of the generators \(\gamma_ 1,...,\gamma_{2g}\) and any lifts \({\hat \gamma}{}_ 1,...,{\hat \gamma}_{2g}\) to SL(2,\({\mathbb{R}})\), \(\prod_{i odd}[{\hat \gamma}_ i,{\hat \gamma}_{i+1}]=I.\)
Reviewer: M.Engber

MSC:
30F10 Compact Riemann surfaces and uniformization
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
55Q05 Homotopy groups, general; sets of homotopy classes