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Traces of commutators of Möbius transformations. (English) Zbl 0898.30037

The question of lifting subgroups of \(\text{PSL}(2,\mathbb{R})\) to \(\text{SL}(2,\mathbb{R})\) has an interesting history. In this connection see papers by the reviewer in [Differential geometry and complex analysis, Vol. dedic. H. E. Rauch, 181-193 (1985; Zbl 0571.30037)] and by M. Culler [Adv. Math. 59, No. 1, 64-70 (1986; Zbl 0582.57001)]. The authors reprove by elementary methods, that a Fuchsian group uniforming a compact surface of genus \(>1\) lifts. The basic fact established in this paper is that the trace of the commutator of a pair of hyperbolic Möbius transformations with intersecting axes is negative.

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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