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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: October 25, 2007
MathSciNet review: 2358474
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Abstract: A relative one-relator presentation has the form $ \mathcal{P} = \langle \mathbf{x}, H; R \rangle$ where $ \mathbf{x}$ is a set, $ H$ is a group, and $ R$ is a word on $ \mathbf{x}^{\pm 1} \cup H$. We show that if the word on $ \mathbf{x}^{\pm 1}$ obtained from $ R$ by deleting all the terms from $ H$ has what we call the unique max-min property, then the group defined by $ \mathcal{P}$ is residually finite if and only if $ H$ 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 $ \mathcal{P}$ (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: http://dx.doi.org/10.1090/S0002-9939-07-09160-5
PII: S 0002-9939(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.