Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T13:33:20.257Z Has data issue: false hasContentIssue false

Geometry of Neumann subgroups

Published online by Cambridge University Press:  09 April 2009

Ravi S. Kulkarni
Affiliation:
Department of Mathematics, City University of New York33 W. 42nd Street New York, New York 10036–8099, U.S.A.
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.

A Neumann subgroup of the classical modular group is by definition a complement of a maximal parabolic subgroup. Recently Neumann subgroups have been studied in a series of papers by Brenner and Lyndon. There is a natural extension of the notion of a Neumann subgroup in the context of any finitely generated Fuchsian group Γ acting on the hyperbolic plane H such that Γ/H is homeomorphic to an open disk. Using a new geometric method we extend the work of Brenner and Lyndon in this more general context.

MSC classification

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Brenner, J. L. and Lyndon, R. C., ‘Nonparabolic subgroups of the modular group’, J. Algebra 77 (1982), 311322.CrossRefGoogle Scholar
[2]Brenner, J. L. and Lyndon, R. C., ‘Permutations and cubic graphs’, Pacific J. Math. 104 (1983), 285315.CrossRefGoogle Scholar
[3]Brenner, J. L. and Lyndon, R. C., ‘Maximal nonparabolic subgroups of the modular group’, Math. Ann. 263 (1983), 111.CrossRefGoogle Scholar
[4]Kulkarni, R. S., ‘An extension of a theorem of Kurosh and applications to Fuchsian groups’, Michigan Math. J. 30 (1983), 259272.CrossRefGoogle Scholar
[5]Kulkarni, R. S., ‘Geometry of free products’, Math. Z. 193 (1986), 613624.CrossRefGoogle Scholar
[6]Magnus, W., ‘Rational representations of Fuchsian groups and nonparabolic subgroups of the modular group’, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 9 (1973), 179189.Google Scholar
[7]Magnus, W., Noneuclidean tesselations and their groups, (Academic Press, New York, 1974).Google Scholar
[8]Neumann, B. H., ‘Über ein gruppen-theoretisch-arithmetisches problem’, Sitzungsber. Preuss. Akad. Wiss. Math.-Phys. Kl. 10 (1933).Google Scholar
[9]Petersson, H., ‘Über einen einfachen typus von untergurppen der modulgruppe’, Arch. Math. 4 (1953), 308315.CrossRefGoogle Scholar
[10]Stothers, W. W., ‘Subgroups of infinite index in the modular group I-III’, Glasgow Math. J. 20 (1979), 103114.CrossRefGoogle Scholar
Glasgow Math. J.. 22 (1981), 101118,.CrossRefGoogle Scholar
Glasgow Math. J. 22 (1981), 119131.CrossRefGoogle Scholar
[13]Tretkoff, C., ‘Nonparabolic subgroups of the modular group’, Glasgow Math. J. 16 (1975), 90102.CrossRefGoogle Scholar