Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-24T17:45:47.597Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounding finite groups acting on 3-manifolds

Published online by Cambridge University Press:  24 October 2008

Sadayoshi Kojima
Sadayoshi Kojima
Affiliation:
Department of Mathematics, Tokyo Metropolitan University, Japan
Get access Rights & Permissions [Opens in a new window]

Extract

In Problem 3·39 (B) and (C) of Kirby's collection [10], Giffen and Thurston asked whether, for a closed 3-manifold M, the order of finite subgroups of Diff M is bounded, so that it contains no infinite torsion subgroups unless M admits a circle action. In this paper, we answer this question affirmatively for homotopy geometric manifolds, and then discuss some hyperbolic 3-manifolds with only a few actions as examples showing poor symmetry of 3-manifolds in general.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 2 , September 1984 , pp. 269 - 281
Copyright
Copyright © Cambridge Philosophical Society 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Borel, A.. Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Scuola Norm. Sup. Pisa CI. Sci. (41) 8 (1981), 1–34.Google Scholar
[2]Evans, B. and Moser, L.. Solvable fundamental groups of compact 3-manifolds. Trans. Amer. Math. Soc. 168 (1972), 189–210.CrossRefGoogle Scholar
[3]Freedman, M. and Yau, S. T.. Homotopically trivial symmetries of Haken manifolds are toral. Topology 22 (1983), 179–189.CrossRefGoogle Scholar
[4]Hartley, R.. Knots with free period. Canad. J. Math. 33 (1981), 91–102.CrossRefGoogle Scholar
[5]Hartley, R.. Identifying non-invertible knots. Topology 22 (1983), 137–143.CrossRefGoogle Scholar
[6]Jaco, W. and Shalen, P.. Seifert fibered spaces in 3-manifolds. Mem. Amer. Math. Soc. no. 220 (1979)Google Scholar
[7]Johannson, K.. Homotopy Equivalences of 3-manifolds with Boundaries. Lecture Notes in Math. no. 761 (1979).CrossRefGoogle Scholar
[8]Kawauchi, A.. The invertibility problem for amphicheiral excellent knots. Proc. Japan Acad. 55 (1979), 399–402.Google Scholar
[9]Kerckhoff, S.. The Nielsen realization problem. Ann. of Math. (2) 117 (1983), 235–265.CrossRefGoogle Scholar
[10]Kirby, R.. Problems in low-dimensional manifold theory. Proc. Sympos. Pure Math. 32 (1978), 273–312.CrossRefGoogle Scholar
[11]Kobayashi, T.. Equivariant annulus theorem for 3-manifolds. Proc. Japan Acad. 59 (1983), 403–406.Google Scholar
[12]Kojima, S.. Finiteness of symmetries on 3-manifolds (Preprint, 1983).Google Scholar
[13]Mostow, G.. Strong Rigidity for Locally Symmetric Spaces. Ann. of Math. Study 78 (Princeton University Press, 1973).Google Scholar
[14]Meeks, W., Simon, L. and Yau, S. T.. Embedding minimal surfaces, exiotic spheres and manifolds with positive Ricci curvature. Ann. of Math. (2) 116 (1982), 621–659.Google Scholar
[15]Murasugi, K.. On symmetry of knots. Tsukuba J. Math. 4 (1980), 331–347.CrossRefGoogle Scholar
[16]Plotnick, S.. Finite group actions and non-separating 2-spheres (Preprint).Google Scholar
[17]Raymond, F. and Tollefson, J.. Closed 3-manifolds with no periodic maps. Trans. Amer. Math. Soc. 221 (1976), 403–418.CrossRefGoogle Scholar
Corrections to this paper, Trans. Amer. Math. Soc. 272 (1982), 803–807.Google Scholar
[18]Raymond, F.. Classification of the actions of the circle on 3-manifolds. Trans. Amer. Math. Soc. 131 (1968), 51–78.CrossRefGoogle Scholar
[19]Rolfsen, D.. Knots and Links. Math. Lect. Series 7 (Berkeley, Publish or Perish Inc., 1976).Google Scholar
[20]Sakuma, M.. Involutions on torus bundles over S 1 (Preprint, 1984).Google Scholar
[21]Scott, P.. There are no fake Seifert fibered spaces with infinite -π1. Ann. of Math. (2) 117 (1983), 35–70.CrossRefGoogle Scholar
[22]Scott, P.. The geometries of 3-manifolds. Bull. London Math. Soc. 15 (1983), 401–487.CrossRefGoogle Scholar
[23]Scott, P.. Finite group actions on 3-manifolds (Preprint).Google Scholar
[24]Siebenmann, L.. On Vanishing of the Rohlin Invariant and Nonfinitely Amphicheiral Homology 3-spheres. Lecture Notes in Math. no. 788 (1980), 177–222.Google Scholar
[25]Thurston, W.. Three Dimensional Geometry and Topology. To appear in Lecture Note Series (Princeton University Press).Google Scholar
[26]Thurston, W.. Three-manifolds with Symmetry (Preliminary report, 1982).Google Scholar
[27]Waldhausen, K.. On irreducible 3-manifolds which are sufficiently large. Ann. of Math. (2) 87 (1968), 56–88.CrossRefGoogle Scholar
[28] Proceedings of the 1979 Conference on the Smith conjecture at Columbia University (To appear).Google Scholar