On the residual finiteness and other properties of (relative) one-relator groups

Author:
Stephen J. Pride

Journal:
Proc. Amer. Math. Soc. **136** (2008), 377-386

MSC (2000):
Primary 20E26, 20F05; Secondary 20F10, 57M07

DOI:
https://doi.org/10.1090/S0002-9939-07-09160-5

Published electronically:
October 25, 2007

MathSciNet review:
2358474

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Abstract: A relative one-relator presentation has the form where is a set, is a group, and is a word on . We show that if the word on obtained from by deleting all the terms from has what we call the *unique max-min property*, then the group defined by is residually finite if and only if is residually finite (Theorem 1). We apply this to obtain new results concerning the residual finiteness of (ordinary) one-relator groups (Theorem 4). We also obtain results concerning the conjugacy problem for one-relator groups (Theorem 5), and results concerning the relative asphericity of presentations of the form (Theorem 6).

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Additional Information

**Stephen J. Pride**

Affiliation:
Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, Scotland, United Kingdom

Email:
sjp@maths.gla.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-07-09160-5

Keywords:
Residual finiteness,
one-relator group,
relative presentation,
(power) conjugacy problem,
asphericity,
unique max-min property,
2-complex of groups,
covering complex

Received by editor(s):
June 5, 2006

Published electronically:
October 25, 2007

Communicated by:
Jonathan I. Hall

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.