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On Torsion-Free Discrete Subgroups of PSL(2, C) with Compact Orbit Space

Published online by Cambridge University Press:  20 November 2018

L. A. Best
L. A. Best*
Affiliation:
The University of Birmingham, Birmingham, England
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In 1883 Poincaré [13] recognized that the discrete subgroups of PSL(2, C) could be extended from their natural action on the complex plane to acting on hyperbolic 3-space and he attempted to analyze these groups in an analogous manner to his classical treatment of Fuchsian groups, with fundamental polyhedra playing the role of the fundamental polygons for Fuchsian groups. This approach, however, did not lead very far, perhaps not surprisingly when one appreciates the close connection between the geometry of these groups and the topology of 3-manifolds. Since that time the state of knowledge remained essentially unchanged until 1964 when work by Ahlfors [1] and soon afterwards by Bers [3] revitalized the subject of Kleinian groups. The modern approach tends to use analytic methods, although recently Marden [11] has had considerable success in carrying forward Poincaré's geometric approach.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 451 - 460
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Ahlfors, L. V., Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), 413–429.Google Scholar
2. Armstrong, M. A., On the fundamental group of an orbit space, Proc. Cambridge Philos. Soc. 61 (1965), 639–646.Google Scholar
3. Bers, L., Inequalities for finitely generated Kleinian groups, J. Analyse Math. 18 (1967), 23–41.Google Scholar
4. Coxeter, H. S. M., Regular honeycombs in hyperbolic space, Proc. Internat. Congr. Math., 1954, Amsterdam, Vol. III, pp. 155–169 (Noordhoff, Gröningen; North-Holland, Amsterdam, 1956).Google Scholar
5. Greenberg, L., Fundamental polyhedra for Kleinian groups, Ann. of Math. (2) 84 (1966), 433–441.Google Scholar
6. Hall, M., Jr., The theory of groups (Macmillan, New York, 1959).Google Scholar
7. Lannér, F., On complexes with transitive groups of automorphisms, Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 11 (1950), 41 pp.Google Scholar
8. Macbeath, A. M., Discontinuous groups and birational transformations, Dundee Summer School in Geometry and Topology (1961) (Queen's College, Dundee, Scotland; University of St. Andrews, St. Andrews, Scotland, 1961, also 1965).Google Scholar
9. Macbeath, A. M., Groups of homeomorphisms of a simply connected space, Ann. of Math. (2) 79 (1964), 473–488.Google Scholar
10. Macbeath, A. M. and Singerman, D., Spaces of subgroups of Lie groups (to appear).Google Scholar
11. Marden, A., The geometry of finitely generated Kleinian groups without torsion (to appear).Google Scholar
12. Maskit, B., On Poincaré's theorem for fundamental polygons (to appear).Google Scholar
13. Poincaré, H., Oeuvres de Henri Poincaré, Publiées sous les auspices de l'Académie des Sciences, Vol. 2, pp. 258-299, Mémoire sur les groupes Kleinêens (Gauthier-Villars, Paris, 1916); Acta Math. S (1883), 49–92.Google Scholar
14. Reidemeister, K., Einfûhrung in die Kombinatorische Topologie (F. Vieweg, Braunschweig, 1932).Google Scholar
15. Weber, C. and Seifert, H., Die beiden Dodekaederràume, Math. Z. 37 (1933), 237–253.Google Scholar