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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