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A note on centrality in 3-manifold groups

Published online by Cambridge University Press:  24 October 2008

P. H. Kropholler
P. H. Kropholler
Affiliation:
School of Mathematical Sciences, Queen Mary College, London E1 4NS
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Centralizers in fundamental groups of 3-manifolds are well understood because of their relationship with Seifert fibre spaces. Jaco and Shalen's book [4] provides detailed information about Seifert fibre spaces in 3-manifolds, and consequently about centralizers in their fundamental groups. It is the purpose of this note to record two group-theoretic properties, both easily deduced from results of Jaco and Shalen. Doubtless many other authors could have established the same results had they needed them. Our motivation for writing this paper is that these properties can be used as a basis for group-theoretic proofs of certain fundamental results in 3-manifold theory: Proposition 1 below can be used as a basis for a proof of the Torus Theorem (cf. [8]) and Proposition 2 for the Torus Decomposition Theorem (cf. [9]).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 2 , March 1990 , pp. 261 - 266
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Bieri, R.. Homological Dimension of Discrete Groups, 2nd ed.Queen Mary College Math. Notes (1981).Google Scholar
[2]Hempel, J.. 3-Manifolds. Ann. of Math. Stud. no. 86 (Princeton University Press, 1976).Google Scholar
[3]Hillman, J. A.. Three-dimensional Poincaré duality groups which are extensions. Math. Z. 195 (1987), 89–92.CrossRefGoogle Scholar
[4]Jaco, W. H. and Shalen, P. B.. Seifert Fibered Spaces in 3-Manifolds. Mem. Amer. Math. Soc. vol. 21 no. 220 (1979).Google Scholar
[5]Kawauchi, A.. A test for the fundamental group of a 3-manifold. J. Pure Appl. Algebra 28 (1983), 189–196.CrossRefGoogle Scholar
[6]Kropholler, P. H. and Roller, M. A.. Splittings of Poincaré duality groups. Math. Z. 197 (1988), 421–438.CrossRefGoogle Scholar
[7]Kropholler, P. H. and Roller, M. A.. Splittings of Poincaré duality groups III. J. London Math. Soc. 39 (1989), 271–284.CrossRefGoogle Scholar
[8]Kropholler, P. H.. Amalgams, HNN-extensions and the Torus Theorem. (In preparation.)Google Scholar
[9]Kropholler, P. H.. An analogue of the torus decomposition theorem for certain Poincaré duality groups. Proc. London Math. Soc. (to appear).Google Scholar
[10]Milnor, J.. Groups which act on Sn without fixed points. Amer. J. Math. 79 (1957), 623–630.CrossRefGoogle Scholar
[11]Passman, D. S.. The Algebraic Structure of Group Rings (Wiley-Interscience, 1977).Google Scholar
[12]Scott, P.. A new proof of the annulus and torus theorems. Amer. J. Math. 102 (1980), 241–277.CrossRefGoogle Scholar
[13]Scott, P.. Finitely generated 3-manifold groups are finitely presented. J. London Math. Soc. (2) 6 (1972), 437–440.Google Scholar
[14]Scott, P.. Compact submanifolds of 3-manifolds. J. London Math. Soc. (2) 7 (1973), 246–250.CrossRefGoogle Scholar