Skip to main content Accessibility help
×
Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-19T02:08:10.687Z Has data issue: false hasContentIssue false

On perfect subgroups of one-relator groups

Published online by Cambridge University Press:  05 April 2013

J Harlander
Affiliation:
Universität Frankfurt
Andrew J. Duncan
Affiliation:
University of Newcastle upon Tyne
N. D. Gilbert
Affiliation:
University of Durham
James Howie
Affiliation:
Heriot-Watt University, Edinburgh
Get access

Summary

If P is a group of operators on a group A, then we denote by dp(A) the minimal number of generators of A as a P-group.

Suppose G is a group and F/N is a presentation of G. Then F acts on N by conjugation and induces an action of G on Nab. This ℤG-module is called the relation module of the presentation F/N.

Definition. A presentation F/N of a group G is said to have a relation gap if dG(Nab) is strictly less than dF(N).

It is an open problem whether there exists a presentation that has a relation gap (see Harlander [H1], [H2] and Baik, Pride [B-P]). Such a presentation would be interesting not only to group theorists. In [D] Dyer shows that a presentation with a relation gap could be used to settle an open question concerning complexes dominated by a 2-complex (see also Wall [W] and Ratcliffe [R]).

In [H1] and [H2] the author studies groups that have cyclic relation modules. Such groups are quotients G/P, with G a one-relator group, say presented by 〈Xr〉, and P a perfect normal subgroup G. Now if P is not of the form 〈wF / 〈rF, where 〈wF denotes the normal closure of the element w of the free group F on X, then G/P has a presentation with a relation gap.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] 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.

Available formats
×

Save book to Dropbox

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

Available formats
×

Save book to Google Drive

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

Available formats
×