Skip to main content Accessibility help
×
Home
Hostname: page-component-7f7b94f6bd-5bz6h Total loading time: 0.177 Render date: 2022-06-29T15:22:39.976Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Remarks on the Intersection of Finitely Generated Subgroups of a Free Group

Published online by Cambridge University Press:  20 November 2018

R. G. Burns
Affiliation:
R. G. Burns, Department Of Mathematics, York University, North York, Toronto, Ontario, CanadaM3J 1P3
Wilfried Imrich
Affiliation:
Wilfried Imrich, Institute For Mathematics and Applied Geometry, Montanuniversität Leoben, A-8700, Leoben, Austria
Brigitte Servatius
Affiliation:
Brigitte Servatius, Department of Mathematics, Syracuse University, Syracuse, N.Y. 13210, U.S.A.
Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The first result gives a (modest) improvement of the best general bound known to date for the rank of the intersection U ∩ V of two finite-rank subgroups of a free group F in terms of the ranks of U and V. In the second result it is deduced from that bound that if A is a finite-rank subgroup of F and B < F is non-cyclic, then the index of A ∩ B in B, if finite, is less than 2(rank(A) - 1), whence in particular if rank (A) = 2, then B ≤ A. (This strengthens a lemma of Gersten.) Finally a short proof is given of Stallings' result that if U, V (as above) are such that U ∩ V has finite index in both U and V, then it has finite index in their join 〈U, V〉.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Bums, Robert G., On the intersection of finitely generated subgroups of a free group, Math. Zeitschr. 119(1971), pp. 121130.Google Scholar
2. Bums, R. G., A note on free groups, Proc. Amer. Math. Soc. 23 (1969), pp. 14—17.Google Scholar
3. Gersten, S.M., Intersections of finitely generated subgroups of free groups and resolutions of graphs, Invent. Math. 71 (1983), pp. 567591.CrossRefGoogle Scholar
4. Greenberg, L., Discrete groups of motions, Canad. J. Math. 12 (1960), pp. 414425.CrossRefGoogle Scholar
5. Howson, A.G., On the intersection of finitely generated free groups, J. London Math. Soc. 29 (1954), pp. 428434.CrossRefGoogle Scholar
6. Karrass, A. and Solitar, D., On finitely generated subgroups of a free group, Proc. Amer. Math. Soc. 22 (1969), pp. 209213.CrossRefGoogle Scholar
7. Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory, (Interscience, New York, 1966).Google Scholar
8. Neumann, Hanna, On the intersection of finitely generated free groups, Publ. Math. Debrecen 4 (1956), 186189. Addendum, Publ. Math. Debrecen 5 (1957/58), p. 128.Google Scholar
9. Nickolas, Peter, Intersections of finitely generated free groups. Bull. Austral. Math. Soc, 31 (1985), pp. 339348.CrossRefGoogle Scholar
10. Servatius, Brigitte, A short proof of a theorem of Burns, Math. Zeitschr. 184 (1983), pp. 133—137.CrossRefGoogle Scholar
11. Stallings, John R., Topology of finite graphs, Invent. Math. 71 (1983), pp. 551565.CrossRefGoogle Scholar
You have Access
7
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Remarks on the Intersection of Finitely Generated Subgroups of a Free Group
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Remarks on the Intersection of Finitely Generated Subgroups of a Free Group
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Remarks on the Intersection of Finitely Generated Subgroups of a Free Group
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *