The length and thickness of words

in a free group

Author:
R. Z. Goldstein

Journal:
Proc. Amer. Math. Soc. **127** (1999), 2857-2863

MSC (1991):
Primary 20E05

Published electronically:
May 4, 1999

MathSciNet review:
1641693

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Abstract | References | Similar Articles | Additional Information

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 , then any word equivalent to it in a free group of rank has length at least . 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.

**1.**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**, 10.1007/BF01388713**2.**Abe Shenitzer,*Decomposition of a group with a single defining relation into a free product*, Proc. Amer. Math. Soc.**6**(1955), 273–279. MR**0069174**, 10.1090/S0002-9939-1955-0069174-1**3.**J. Singer,*Three Dimensional Manifolds and Their Heegaard Diagrams*, Trans Amer. Math. Soc.**35**(1933), 88-111.**4.**J. H. C. Whitehead,*On Certain Sets of Elements in a Free Group*, Proc. London Math. Soc.**41**(1936), 48-56.**5.**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

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-05142-4

Received by editor(s):
January 11, 1998

Published electronically:
May 4, 1999

Communicated by:
Ronald M. Solomon

Article copyright:
© Copyright 1999
American Mathematical Society