Convergence groups are Fuchsian groups. (English) Zbl 0733.57022
This paper presents a sketch of a complete solution of the well-known problem about discrete groups of homeomorphisms of the circle \(S^ 1\) with a convergence property, i.e., the convergence groups [see F. Gehring and G. Martin, Proc. Lond. Math. Soc., III. Ser. 55, 331- 358 (1987; Zbl 0628.30027)]. Namely, the author proves that any such group is conjugate in \(Homeo(S^ 1)\) to the restriction of a Fuchsian group in the unit disc B, \(\partial B=S^ 1\). This gives a new proof of the Nielsen realization problem [S. Kerckhoff, Ann. Math., II. Ser. 117, 235-265 (1983; Zbl 0528.57008)] and completes arguments of G. Mess [The Seifert conjecture and groups which are coarse quasi isometric to planes, Preprint] and P. Scott [Ann. Math., II. Ser. 117, 35-70 (1983; Zbl 0516.57006)] giving a proof of the Seifert fiber space conjecture:
Corollary. Let M be a compact, orientable, irreducible (i.e. every smooth embedded \(S^ 2\) bounds a 3-cell) 3-manifold with infinite \(\pi_ 1\), then M is a Seifert fibered space if and only if \(\pi_ 1(M)\) contains a cyclic normal subgroup.
The other corollary is the Torus Theorem [see C. D. Feustel, Trans. Am. Math. Soc. 217, 1-43, 45-57 (1976; Zbl 0321.55006 and Zbl 0321.55007); and P. Scott, Am. J. Math. 102, 241-277 (1980; Zbl 0439.57004)]: If M is an orientable, irreducible 3-manifold and \(Z\oplus Z\subset \pi_ 1(M)\), then M is either Seifert fibered or contains an incompressible torus (i.e. an embedded torus whose induced \(\pi_ 1\) map is injective).
We remark that A. Casson and D. Jungreis [Convergence groups and Seifert fibered 3-manifolds, Preprint] have given a completely different proof of the main theorem. Also P. Tukia [J. Reine Angew. Math. 391, 1-54 (1988; Zbl 0644.30027)] proved the main result for convergence groups G without torsion elements of order greater than 3 and also in the case of non-compact action of G.
Corollary. Let M be a compact, orientable, irreducible (i.e. every smooth embedded \(S^ 2\) bounds a 3-cell) 3-manifold with infinite \(\pi_ 1\), then M is a Seifert fibered space if and only if \(\pi_ 1(M)\) contains a cyclic normal subgroup.
The other corollary is the Torus Theorem [see C. D. Feustel, Trans. Am. Math. Soc. 217, 1-43, 45-57 (1976; Zbl 0321.55006 and Zbl 0321.55007); and P. Scott, Am. J. Math. 102, 241-277 (1980; Zbl 0439.57004)]: If M is an orientable, irreducible 3-manifold and \(Z\oplus Z\subset \pi_ 1(M)\), then M is either Seifert fibered or contains an incompressible torus (i.e. an embedded torus whose induced \(\pi_ 1\) map is injective).
We remark that A. Casson and D. Jungreis [Convergence groups and Seifert fibered 3-manifolds, Preprint] have given a completely different proof of the main theorem. Also P. Tukia [J. Reine Angew. Math. 391, 1-54 (1988; Zbl 0644.30027)] proved the main result for convergence groups G without torsion elements of order greater than 3 and also in the case of non-compact action of G.
Reviewer: B.N.Apanasov (Norman)
MSC:
| 57S25 | Groups acting on specific manifolds |
| 57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |
| 57M05 | Fundamental group, presentations, free differential calculus |
| 20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |
| 57N10 | Topology of general \(3\)-manifolds (MSC2010) |
Keywords:
discrete groups of homeomorphisms; Fuchsian group; Nielsen realization problem; Seifert fiber space; Torus TheoremCitations:
Zbl 0628.30027; Zbl 0528.57008; Zbl 0516.57006; Zbl 0321.55006; Zbl 0321.55007; Zbl 0439.57004; Zbl 0644.30027References:
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