On the divisibility of knot groups

Abstract

A group G is called an R-group if for every pair of elements x and y, and every natural number nf it follows from x n = y that x = y. In other words, G is an ϋϊ-group if G has not more than one solution for every equation x = a. If G is an iϋ-group, G is locally infinite. The converse, however, need not be true even if G is restricted to the group of a knot in S. For example, let G be the group of K(m, ri), the torus knot of type (m, n). G has a presentation G = (α, b: a = b). Then the equation x = a has infinitely many distinct solutions, x = a, (ba)a(ba)ιf (ba) a(ba)~, This observation gives immediately a negative answer to Problem N in [3]. Neuwirth asks if a knot group can be ordered. In fact, the group of K{m, n)( m |, | n | ^ 2) cannot be ordered, since an ordered group is always an i?-group. Therefore, Problem N now leads slightly weaker problems: Can a knot group other than torus knot groups be ordered? Or, is a knot group other than torus knot groups an iZ-group? The purpose of this paper is to give a sufficient condition for the group of a fibred knot to be an iϋ-group. (See Theorem 2.) Using this condition, we can prove, for example, that the group of the figure eight knot is an i2-group. (See Proposition 3 or Proposition 5.)