Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T00:30:08.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Totally geodesic surfaces in hyperbolic 3-manifolds

Published online by Cambridge University Press:  20 January 2009

Alan W. Reid
Alan W. Reid
Affiliation:
Department of MathematicsThe Ohio State University231 West 18th AvenueColumbus, OH 43210, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we investigate totally geodesic surfaces in hyperbolic 3-manifolds. In particular we show that if M is a compact arithmetic hyperbolic 3-manifold containing an immersion of a totally geodesic surface then it contains infinitely many commensurability classes of such surfaces. In addition we show for these M that the Chern-Simons invariant is rational.

We also show, that unlike the figure-eight knot complement in S3, many knot complements in S3 do not contain an immersion of a closed totally geodesic surface.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 1 , February 1991 , pp. 77 - 88
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

REFERENCES

1.Adams, C. C., Thrice punctured spheres in hyperbolic 3-manifolds, Trans. Amer. Math. Soc. 287 (1985), 645–656.Google Scholar
2.Adams, C. C., Augmented alternating link complements are hyperbolic, in Low-dimensional Topology and Kleinian groups (ed. Epstein, D. B. A., LMS Lecture Note Series 112 (1986), 115–130.Google Scholar
3.Borel, A., Commensurability classes and volumes of hyperbolic 3-manifolds, Ann. Scuola Norm. Sup. Pisa 8 (1931), 1–33.Google Scholar
4.Burde, G. and Zieschang, H., Knots (De Gruyter Studies in Math. 5. Walter De Gruyter, Berlin, New York, 1985).Google Scholar
5.Chinburg, T., A small arithmetic hyperbolic 3-manifold. Proc. Amer. Math. Soc. 100 (1987), 140–144.CrossRefGoogle Scholar
6.Clozel, L., On the cuspidal cohomology of arithmetic subgroups of SL (2n) and the first betti number of arithmetic 3-manifolds, Duke Math. J. 55 (1987), 475–486.Google Scholar
7.Frohman, C. and Fine, B., Some amalgam structures for Bianchi groups, Proc. Amer. Math. Soc. 102 (1988), 221–229.Google Scholar
8.Goldstein, L. J., Analytic Number Theory (Prentice-Hall Inc., New Jersey, 1971).Google Scholar
9.Gordon, C. M. and Litherland, R. A., Incompressible surfaces in branched coverings, in The Smith Conjecture (eds. Morgan, J. W. and Bass, H., Academic Press, 1984). 139–152.Google Scholar
10.Greenberg, L., On a theorem of Ahlfors and conjugate subgroups of Kleinian groups, Amer. J. Math. 89 (1967), 55–68.Google Scholar
11.Long, D. D., Immersions and embeddings of totally geodesic surfaces, Bull. London Math. Soc. 19 (1987), 481–484.Google Scholar
12.Macbeath, A. M., Commensurability of cocompact three dimensional hyperbolic groups, Duke Math. J. 50 (1983), 1245–1253.Google Scholar
13.Maclachlan, C., Fuchsian subgroups of the groups PSL 2(0d), in Low-dimensional Topology and Kleinian groups, ed. Epstein, D. B. A., L. M. S. Lecture Note Series 112 (1986), 305–311.Google Scholar
14.Maclachlan, C. and Reid, A. W., Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups, Math. Proc. Cambridge Philos. Soc. 102 (1987), 251–257.Google Scholar
15.Maclachlan, C. and Reid, A. W., In preparation.Google Scholar
16.Meyerhoff, R., Density of the Chern-Simons' invaritant for hyperbolic 3-manifolds, in Low-dimensional Topology and Kleinian groups ed. Epstein, D. B. A., L. M. S. Lecture Note Series 112 (1986), 217–239.Google Scholar
17.Reid, A. W., Ph.d thesis, University of Aberdeen, 1987.Google Scholar
18.Reid, A. W., Arithmeticity of knot complements, J. London Math. Soc., to appear.Google Scholar
19.Reid, A. W., A note on trace-fields of Kleinian groups, Bull. London Math. Soc., to appear.Google Scholar
20.Riley, R., Parabolic representations of knot groups, I, Proc. London Math. Soc. (3) 24 (1972), 217–247.Google Scholar
21.Riley, R., Knots with parabolic property P, Quart. J. Math. Oxford (2) 25 (1974), 273–283.CrossRefGoogle Scholar
22.Riley, R., A quadratic parabolic group, Math. Proc. Cambridge Philos. Soc. 77 (1975), 281–288.Google Scholar
23.Riley, R., Seven excellent knots, in Low-dimensional topology (ed Brown, S. R. and Thickstun, T. L., L. M. S. Lecture Note Series 48 (1982), 81–151.Google Scholar
24.Riley, R., Parabolic representations and symmetries of the knot 932 in Computers in Geometry and Topology (ed Tangora, D. M. C., Lecture notes in Pure and Applied Math. 114 (1988), 297–313.Google Scholar
25.Takeuchi, K., On some discrete subgroups of SL 2(R). J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 16 (1969), 97–100.Google Scholar
26.Takeuchi, K., A characterization of arithmetic Fuchsian groups. J. Math. Soc. Japan 27 (1975), 600–612.Google Scholar
27.Takeuchi, K., Arithmetic triangle groups, J. Math. Soc. Japan 29 (1977), 91–106.Google Scholar
28.Takeuchi, K., Commensurability classes of arithmetic triangle groups, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 24 (1977), 201–212.Google Scholar
29.Thurston, W. P., The geometry and topology of 3-manifolds (Mimeographed lecture notes, Princeton Univ., 1978).Google Scholar
30.Vigneras, M.-F., Arithmétique des algèbres de quaternions (Lecture Notes in Mathematics 800, Springer-Verlag, 1980).Google Scholar
31.Wielenberg, N. J., The structure of certain subgroups of the Picard group, Math. Proc. Cambridge Philos. Soc. 84 (1978), 427–436.Google Scholar