The length and thickness of words in a free group

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 t , then any word equivalent to it in a free group of rank n n has length at least 2 n t 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.