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Intersections of conjugates of Magnus subgroups of one-relator groups

Donald J Collins

Geometry & Topology Monographs 14 (2008) 135–171

DOI: 10.2140/gtm.2008.14.135

arXiv: 0904.1358


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.


one-relator group, Magnus subgroup

Mathematical Subject Classification

Primary: 20F05



Received: 14 March 2006
Accepted: 30 January 2007
Published: 29 April 2008

Donald J Collins
School of Mathematical Sciences
Queen Mary, University of London
Mile End Road
London E1 4NS