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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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The length and thickness of words in a free group
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by R. Z. Goldstein PDF
Proc. Amer. Math. Soc. 127 (1999), 2857-2863 Request permission

Abstract:

In this paper we generalize the notion of a cut point of a graph. We assign to each graph a non-negative integer, called its thickness, so that a graph has thickness 0 if and only if it has a cut point. We then apply a method of J. H. C. Whitehead to show that if the coinitial graph of a given word has thickness $t$, then any word equivalent to it in a free group of rank $n$ has length at least $2nt$. We also define what it means for a word in a free group to be separable and we show that there is an algorithm to decide whether or not a given word is separable.
References
  • Richard Z. Goldstein and Edward C. Turner, Automorphisms of free groups and their fixed points, Invent. Math. 78 (1984), no. 1, 1–12. MR 762352, DOI 10.1007/BF01388713
  • Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
  • J. Singer, Three Dimensional Manifolds and Their Heegaard Diagrams, Trans Amer. Math. Soc. 35 (1933), 88–111.
  • J. H. C. Whitehead, On Certain Sets of Elements in a Free Group, Proc. London Math. Soc. 41 (1936), 48–56.
  • J. H. C. Whitehead, On Equivalent Sets of Elements in a Free Group, Ann. of Math. 37 (1936), 782–800.
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Additional Information
  • R. Z. Goldstein
  • Affiliation: Department of Mathematics, State University at Albany, 1400 Washington Ave., Albany, New York 12222
  • Received by editor(s): January 11, 1998
  • Published electronically: May 4, 1999
  • Communicated by: Ronald M. Solomon
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 2857-2863
  • MSC (1991): Primary 20E05
  • DOI: https://doi.org/10.1090/S0002-9939-99-05142-4
  • MathSciNet review: 1641693