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Homeomorphic conjugates of Fuchsian groups. (English) Zbl 0644.30027
Let $$S=\{x\in R^ 2: | x| =1\}$$. A convergence group of S is a group of homeomorphism of S such that, if $$g_ i\in G$$ are distinct, then it is possible to pass a subsequence in such a way that for some $$x,y\in S$$ the maps $$g_ i| S\setminus \{x\}$$ converge locally uniformly to the constant map $$z\mapsto y$$. This definition is due to F. W. Gehring and G. J.. Martin [Proc. Lond. Math. Soc., III. Ser. 55, 331-358 (1987; Zbl 0628.30027)] and corresponds to their notion of a discrete convergence group.
We conjecture in the paper that a group G of homeomorphisms of the circle S is topologically conjugate to a Fuchsian group (considered to act on S) if and only if it is a convergence group. Although we did not obtain the complete proof, we could prove that either such a group is topologically conjugate to a Fuchsian group, or it contains a finite-index semitriangle subgroup, i.e. a group which is not a finite extension of a cyclic group and which is generated by a and b such that a, b and ab are of finite order. Other conditions implying the topological conjugacy are that G is torsionless, G acts discontinuously somewhere on S, or G is infinitely generated, and many more are given in the paper.
The method is to extend G to act as a convergence group of $$\bar D = \{x\in \bar R^ 2: | x| \leq 1\}$$; when G acts on $$\bar D,$$ the conformal structure of $$\bar D$$ is changed so that G becomes a group of conformal automorphisms of $$\bar D.$$ The extension of the action to $$\bar D$$ is obtained little by little, first extending the action to more and more complicated 1-complexes $$X\subset \bar D$$ and finally to $$\bar D.$$
Our results have relevance on some important problems. One of them, discussed in the paper, is the Nielsen realization problem, recently proved by S. P. Kerckhoff [Ann. Math., II. Ser. 117, 235-265 (1983; Zbl 0528.57008)]; if we could prove our theorem for all groups, we would obtain a new proof of it. Another problem is the Seifert conjecture which gives a characterization of 3-manifolds homeomorphic to Seifert fibered spaces. It has been discussed by G. Mess in “Centers of 3-manifold groups and groups which are coarse quasi-isometric to planes” (to appear).
Reviewer: P.Tukia

##### MSC:
 30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) 57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
##### Keywords:
convergence group; Fuchsian group
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