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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the residual finiteness and other properties of (relative) one-relator groups
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by Stephen J. Pride PDF
Proc. Amer. Math. Soc. 136 (2008), 377-386 Request permission


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:
  • Received by editor(s): June 5, 2006
  • Published electronically: October 25, 2007
  • Communicated by: Jonathan I. Hall
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 377-386
  • MSC (2000): Primary 20E26, 20F05; Secondary 20F10, 57M07
  • DOI:
  • MathSciNet review: 2358474