Lifting surface groups to SL(2,\({\mathbb{C}})\). (English) Zbl 0531.30037

Abikoff, W.; Appel, K.; Schupp, P.

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

Keywords:

fundamental group of compact surface; SL(2,\({\mathbb{R}})\)