Axial pairs and convergence groups on \(S^1\). (English) Zbl 0939.57027

Topology 39, No. 2, 229-237 (2000); erratum ibid. 41, 421-422 (2002).
Let \(G\) be a group of homeomorphisms of \(S^1\). The author proves that \(G\) is a convergence group if it has a set of “axes” which satisfy certain conditions satisfied by Fuchsian groups. The notion of a convergence group was introduced by F. W. Gehring and G. J. Martin [Proc. Lond. Math. Soc., III. Ser. 55, 331-358 (1987; Zbl 0628.30027)]. Convergence groups are topologically conjugate to Fuchsian groups by work of P. Tukia [J. Reine Angew. Math. 391, 1-54 (1988; Zbl 0644.30027)], D. Gabai [Ann. Math., II. Ser. 136, No. 3, 447-510 (1992; Zbl 0785.57004)], and A. Casson and D. Jungreis [Invent. Math. 118, No. 3, 441-456 (1994; Zbl 0840.57005)].
Reviewer: J.Hebda (St.Louis)


57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
11F06 Structure of modular groups and generalizations; arithmetic groups
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)


Fuchsian groups
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