Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-21T09:33:22.515Z Has data issue: false hasContentIssue false
  • English
  • Français

The residual finiteness of HNN-extensions and generalized free products of nilpotent groups: a characterization

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

E. Raptis  and
D. Varsos
E. Raptis
Affiliation:
University of AthensPanepistemiopolis 15784 Athens, Greece
D. Varsos
Affiliation:
University of AthensPanepistemiopolis 15784 Athens, Greece
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.

We study the residual finiteness of free products with amalgamations and HNN-extensions of finitely generated nilpotent groups. We give a characterization in terms of certain conditions satisfied by the associated subgroups. In particular the residual finiteness of these groups implies the possibility of extending the isomorphism of the associated subgroups to an isomorphism of their isolated closures in suitable overgroups of the factors (or the base group in case of HNN-extensions).

MSC classification

Secondary: 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20E26: Residual properties and generalizations; residually finite groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 53 , Issue 3 , December 1992 , pp. 408 - 420
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Andreadakis, S., Raptis, E. and Varsos, D., ‘A characterization of residually finite HNN-extensions of finitely generated abelian groups’, Arch. Math. 50 (1988), 495–501.CrossRefGoogle Scholar
[2]Baumslag, G., Lectures on nilpotent groups, C.B.M.S. Regional Conference Series 2 (Amer. Math. Soc., Providence, R.I., 1971).Google Scholar
[3]Baumslag, G., ‘On the residual finiteness of generalized free products of nilpotent groups’, Trans. Amer. Math. Soc. 106 (1963), 193–209.CrossRefGoogle Scholar
[4]Baumsiag, B., Tretkoff, M., ‘Residually finite HNN-extensions’, Comm. Algebra 6 (1978), 179–194.CrossRefGoogle Scholar
[5]Dyer, J. L., ‘On the residual finiteness of generalized free products’, Trans. Amer. Math. Soc. 133 (1968), 131–143.CrossRefGoogle Scholar
[6]Karras, A., Solitar, D., ‘Subgroups of HNN groups and groups with one defining relation’, Canad. J. Math. 23 (1971), 627–643.CrossRefGoogle Scholar
[7]Kurosh, A. G., The theory of groups (Chelsea, New York 1960).Google Scholar
[8]Shirvani, M., ‘On residually finite HNN-extensions’, Arch. Math. 44 (1985), 110–115.CrossRefGoogle Scholar
[9]Shirvani, M., ‘A converse to a residual finiteness theorem of G. Baumslag’, Proc. Amer. Math. Soc. 104 (1988), 703–706.CrossRefGoogle Scholar