Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-26T13:29:05.347Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-simple geodesics in hyperbolic 3-manifolds

Published online by Cambridge University Press:  24 October 2008

Kerry N. Jones  and
Alan W. Reid
Kerry N. Jones
Affiliation:
Ball State University, Muncie, IN 47306U.S.A.
Alan W. Reid
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

Chinburg and Reid have recently constructed examples of hyperbolic 3-manifolds in which every closed geodesic is simple. These examples are constructed in a highly non-generic way and it is of interest to understand in the general case the geometry of and structure of the set of closed geodesics in hyperbolic 3-manifolds. For hyperbolic 3-manifolds which contain immersed totally geodesic surfaces there are always non-simple closed geodesics. Here we construct examples of manifolds with non-simple closed geodesics and no totally geodesic surfaces.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 2 , September 1994 , pp. 339 - 351
Copyright
Copyright © Cambridge Philosophical Society 1994

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]Bass, H.. Groups of integral representation type. Pacific J. Math. 86 (1980), 15–51.CrossRefGoogle Scholar
[2]Borel, A.. Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Scuola Norm. Sup. Pisa 8 (1981), 1–33.Google Scholar
[3]Chinburg, T. and Reid, A. W.. Closed hyperbolic 3-manifolds whose closed geodesies all are simple. J. Diff. Geom. 38 (1993) 545–558.Google Scholar
[4]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.CrossRefGoogle Scholar
[5]Hass, J.. Surfaces minimizing area in their homology class and group actions on 3-manifolds. Math. Zeit. 199 (1988), 501–509.CrossRefGoogle Scholar
[6]Hilden, H. M., Lozano, M.-T. and Montesinos, J.-M.. A characterization of arithmetic subgroups of SL(2, ℝ) and SL(2, ℂ). Math. Nach. 159 (1992), 245–270.CrossRefGoogle Scholar
[7]Lang, S.. Algebraic Number Theory (Addison Wesley, 1970).Google Scholar
[8]Lam, T. Y.. The Algebraic Theory of Quadratic Forms (Benjamin, 1973).Google Scholar
[9]Maclachlan, C. and Reid, A. W.. Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups. Math. Proc. Cambridge Phil. Soc. 102 (1987), 251–257.CrossRefGoogle Scholar
[10]Neumann, W. D. and Reid, A. W.. Arithmetic of hyperbolic 3-manifolds. TOPOLOGY '90 (Proceedings of the Research Semester in Low Dimensional Topology, Ohio State University) (Apanasov, B., Neumann, W. D., Reid, A. W. and Siebenmann, L., eds.) (De Gruyter, 1992).CrossRefGoogle Scholar
[11]Reid, A. W.. Arithmetic Kleinian groups and their Fuchsian subgroups. Ph.D. thesis, University of Aberdeen, Aberdeen, 1987.Google Scholar
[12]Reid, A. W.. A note on trace-fields of Kleinian groups. Bull. London Math. Soc. 22 (1990), 349–352.CrossRefGoogle Scholar
[13]Reid, A. W.. Totally geodesic surfaces in hyperbolic 3-manifolds. Proc. Edinburgh Math. Soc. 34 (1991), 77–88.CrossRefGoogle Scholar
[14]Takeuchi, K.. Commensurability classes of arithmetic triangle groups. J. Fac. Sci. Univ. Tokyo Sec. IA 24 (1977), 201–212.Google Scholar
[15]Vignéras, M.-F.. Arithmétique des algébres de Quaternions. Lecture Notes in Mathematics 800. (Springer-Verlag, 1980).CrossRefGoogle Scholar