Intersections of conjugates of Magnus subgroups of one-relator groups

Abstract

In the theory of one-relator groups, Magnus subgroups, which are free subgroups obtained by omitting a generator that occurs in the given relator, play an essential structural role. In a previous article, the author proved that if two distinct Magnus subgroups M and N of a one-relator group, with free bases S and T are given, then the intersection of M and N is either the free subgroup P generated by the intersection of S and T or the free product of P with an infinite cyclic group. The main result of this article is that if M and N are Magnus subgroups (not necessarily distinct) of a one-relator group G and g and h are elements of G, then either the intersection of gMg^{-1} and hNh^{-1} is cyclic (and possibly trivial), or gh^{-1} is an element of NM in which case the intersection is a conjugate of the intersection of M and N.