Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T06:49:41.484Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite and infinite cyclic extensions of free groups

Published online by Cambridge University Press:  09 April 2009

A. Karrass ,
A. Pietrowski  and
D. Solitar
A. Karrass
Affiliation:
York UniversityToronto Ontario, Canada
A. Pietrowski
Affiliation:
York UniversityToronto Ontario, Canada
D. Solitar
Affiliation:
York UniversityToronto Ontario, Canada
Rights & Permissions [Opens in a new window]

Extract

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.

Using Stalling's characterization [11] of finitely generated (f. g.) groups with infinitely many ends, and subgroup theorems for generalized free products and HNN groups (see [9], [5], and [7]), we give (in Theorem 1) a n.a.s.c. for a f.g. group to be a finite extension of a free group. Specifically (using the terminology extension of and notation of [5]), a f.g. group G is a finite extension of a free group if and only if G is an HNN group where K is a tree product of a finite number of finite groups (the vertices of K), and each (associated) subgroup Li, Mi is a subgroup of a vertex of K.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 4 , December 1973 , pp. 458 - 466
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Burde, G. and Zieschang, H., ‘Eine Kennzeichung der Torusknoten’, Math. Ann. 169 (1966), 169–176.CrossRefGoogle Scholar
[2]Espstein, D. B. A., Ends, Topology of 3-Manifolds, 110–117, (Prentice-Hall 1962),Google Scholar
[3]Fischer, J., Karrass, A. and Solitar, D., ‘On one-relator groups having elements of finite order’, to appear in Proc. Amer. Math. Soc.Google Scholar
[4]Gregorac, R., ‘On generalized free products of finite extensions of free groups’, J. London Math. Soc. 41 (1966), 662–666.CrossRefGoogle Scholar
[5]Karrass, A. and Solitar, D., ‘The subgroups of a free product of two groups with an anialgamated subgroup’, Trans. Amer. Math. Soc. 150 (1970), 227–255.CrossRefGoogle Scholar
[6]Karrass, A. and Solitar, D., ‘On the free product of two groups with an amalgamated subgroup of finite index in each factor’, Proc. Amer. Math. Soc. 26 (1970), 28–32.CrossRefGoogle Scholar
[7]Karrass, A. and Solitar, D., ‘Subgroups of HNN groups and groups with one defining relation’, Canad. J. Math. 23 (1971), 627–643.CrossRefGoogle Scholar
[8]Karrass, A., Pietrowski, A., and Solitar, D., ‘An improved subgroup theorem for HNN groups’, to be published in Canad. J. Math.Google Scholar
[9]Neumann, Hanna, ‘Generalized free products with amalgamated subgroups’, Amer. J. Math. 70 (1948), 590–625 and 71 (1949), 491–540.CrossRefGoogle Scholar
[10]Pietrowski, A., ‘One-relator groups with center’, submitted for publication.Google Scholar
[11]Stallings, J. R., ‘Groups of cohomological dimension one’, Proc. Symp. Pure Math. XVII (Amer. Math. Soc. 1970), 124–128.Google Scholar
[12]Stallings, J. R., ‘Characterization of tree products of finitely many finite groups’, submitted for publication.Google Scholar