# zbMATH — the first resource for mathematics

Convergence groups and Seifert fibered 3-manifolds. (English) Zbl 0840.57005
The authors give a new proof of D. Gabai’s result, that if $$S^1$$ has a fixed orientation and $$T$$ denotes the set of ordered triples $$(x,y,z)$$ of distinct points occurring in positive order on $$S^1$$, acted on by $$\text{Homeo}_+ (S^1)$$ in the obvious way, and if $$\Gamma \subset \text{Homeo}_+ (S^1)$$ is a discrete convergence group, then $$T/ \Gamma$$ is Seifert fibred. (By work of G. Mess and P. Scott, this implies the Seifert Fibre Space Conjecture.) The proof is by means of braid theory.
Reviewer: C.Kearton (Durham)

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57M60 Group actions on manifolds and cell complexes in low dimensions
##### Keywords:
Seifert fibred; braid; convergence group
Full Text:
##### References:
 [1] [B] R.H. Bing: An alternative proof that 3-manifolds can be triangulated. Ann. Math.69, 37-65 (1959) · Zbl 0106.16604 · doi:10.2307/1970092 [2] [E] D.B.A. Epstein: Periodic flows on 3-manifolds. Ann. Math.95, 66-82 (1972) · Zbl 0231.58009 · doi:10.2307/1970854 [3] [G] D. Gabai: Convergence groups are Fuchsian groups. Ann. Math.136, 447-510 (1992) · Zbl 0785.57004 · doi:10.2307/2946597 [4] [GH] C. Gordon, W. Heil: Cyclic normal subgroups of fundamental groups of 3-manifolds. Topology14, 305-309 (1975) · Zbl 0331.57001 · doi:10.1016/0040-9383(75)90014-2 [5] [GM] F.W. Gehring, G. Martin. Discrete quasiconformal groups I, Proc. London Math. Soc.55, 331-358 (1987) · Zbl 0628.30027 [6] [M] G. Mess: Centers of 3-manifold groups and groups which are coarse quasiisometric to planes (preprint) [7] [S] P. Scott: There are no fake Seifert fibre spaces with infinite ?1. Ann. Math.117, 35-70 (1983) · Zbl 0516.57006 · doi:10.2307/2006970 [8] [T] P. Tukia: Homeomorphic conjugates of Fuchsian groups. J. Reine Angew. Math.391, 1-54 (1988) · Zbl 0644.30027 · doi:10.1515/crll.1988.391.1 [9] [W] F. Waldhausen: On irreducible 3-manifolds which are sufficiently large. Ann. Math.87, 56-88 (1968) · Zbl 0157.30603 · doi:10.2307/1970594
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.