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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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A finitely presented group with unbounded dead-end depth
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by Sean Cleary and Tim R. Riley PDF
Proc. Amer. Math. Soc. 134 (2006), 343-349 Request permission

Erratum: Proc. Amer. Math. Soc. 136 (2008), 2641-2645.


The dead-end depth of an element $g$ of a group $G$, with respect to a generating set $\mathcal {A}$, is the distance from $g$ to the complement of the radius $d_{\mathcal {A}}(1,g)$ closed ball, in the word metric $d_{\mathcal {A}}$ defined with respect to $\mathcal {A}$. We exhibit a finitely presented group $G$ with a finite generating set with respect to which there is no upper bound on the dead-end depth of elements.
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Additional Information
  • Sean Cleary
  • Affiliation: Department of Mathematics, The City College of New York, City University of New York, New York, New York 10031
  • Email:
  • Tim R. Riley
  • Affiliation: Department of Mathematics, Yale University, 10 Hillhouse Avenue, P.O. Box 208283, New Haven, Connecticut 06520-8283
  • Address at time of publication: Department of Mathematics, Cornell University, Ithaca, New York 14853-4201
  • MR Author ID: 691109
  • Email:
  • Received by editor(s): July 26, 2004
  • Received by editor(s) in revised form: September 18, 2004
  • Published electronically: August 12, 2005
  • Additional Notes: Support for the first author from PSC-CUNY grant #65752 is gratefully acknowledged.
    Support for the second author from NSF grant 0404767 is gratefully acknowledged.
  • Communicated by: Alexander N. Dranishnikov
  • © Copyright 2005 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 343-349
  • MSC (2000): Primary 20F65
  • DOI:
  • MathSciNet review: 2176000