Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-04T12:33:03.619Z Has data issue: false hasContentIssue false
  • English
  • Français

Centralizers of reflections in crystallographic groups

Published online by Cambridge University Press:  24 October 2008

T. Baskan  and
A. M. Macbeath
T. Baskan
Affiliation:
Haceteppe University, Ankara, Turkey and University of Pittsburgh
A. M. Macbeath
Affiliation:
Haceteppe University, Ankara, Turkey and University of Pittsburgh
Get access Rights & Permissions [Opens in a new window]

Extract

The study of hyperbolic 3-manifolds has recently been recognized as an increasingly important part of 3-manifold theory (see (9)) and for some time the presence of incompressible surfaces in a 3-manifold has been known to be important (see, for example, (4)). A particularly interesting case occurs when the incompressible surface unfolds in the universal covering space into a hyperbolic plane. The fundamental group of the surface is then contained in the stabilizer of the plane, or, what is the same thing, in the centralizer of the reflection defined by the plane. This is one motivation for studying centralizers of reflections in discrete groups of hyperbolic isometries, or, as we shall call them, hyperbolic crystallographic groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 1 , July 1982 , pp. 79 - 91
Copyright
Copyright © Cambridge Philosophical Society 1982

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)Best, L. A.On torsion-free discrete subgroups of PSL(2, C) with compact orbit-space. Canadian J. of Math. 23 (1971), 451–460.CrossRefGoogle Scholar
(2)Bourbaki, N. Elements de Mathématiques. Groupes et algebres de Lie, ch. 4 (Hermann, Paris, 1968).Google Scholar
(3)Coxeter, H. S. M. and Moser, W. O. J. Generators and relations for discrete groups. Ergebnisse der Math. (Springer, Berlin, 1957).Google Scholar
(4)Hempel, J. 3-manifolds. Annals of Math. Studies, no. 86 (Princeton, 1976).Google Scholar
(5)Lannér, F.On complexes with transitive groups of automorphisms. Comm. Sem. Math. Univ. Lund 11 (1950), 41 pp.Google Scholar
(6)Macbeath, A. M.The classification of non-euclidean crystallographic groups. Canadian J. of Math. 19 (1967), 1192–1205.CrossRefGoogle Scholar
(7)Macbeath, A. M. and Hoare, A. H. M.Groups of hyperbolic crystallography. Math. Proc. Cambridge Phil. Soc. 79 (1976), 235–249.CrossRefGoogle Scholar
(8)Singerman, D.On the structure of non-euclidean crystallographic groups. Math. Proc. Cambridge Phil. Soc. 76 (1974), 233–240.CrossRefGoogle Scholar
(9)Thurston, W. P. The geometry and topology of 3-manifolds (Cyclostyled lecture notes, Princeton, 1977.)Google Scholar
(10)Wilkie, H. C.On non-euclidean crystallographic groups. Math. Zeitechrift 91 (1966), 87–102.CrossRefGoogle Scholar