Homeomorphic conjugates of Fuchsian groups.

*(English)*Zbl 0644.30027Let \(S=\{x\in R^ 2: | x| =1\}\). 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).

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