Intersections of conjugates of Magnus subgroups of one-relator groups

TL;DR

This paper investigates the intersections of conjugates of Magnus subgroups in one-relator groups, revealing conditions under which these intersections are cyclic or conjugate to known intersections, thus deepening understanding of their algebraic structure.

Contribution

The paper extends previous results by characterizing intersections of conjugates of Magnus subgroups, showing they are either cyclic or conjugate to intersections of original subgroups.

Findings

Intersections of conjugates are either cyclic or conjugate to original intersections.

Conditions for when the intersection is trivial, cyclic, or conjugate are established.

Provides a detailed structural description of Magnus subgroup conjugate intersections.

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.