Introduction

Suppose G = 〈x1, …, xn; R = 1〉 is a one-relator group with R a cyclically reduced word in the free group on {x1,…,xn} which involves all the generators. The classical Freiheitssatz or independence theorem of Magnus (see [25]) asserts that the subgroup generated by any proper subset of the generators is free on those generators. More generally suppose X and Y are disjoint sets of generators and suppose that the group A has the presentation A = 〈X; rel(X)〉 and that the group G has the presentation G = 〈X,Y;rel(X),Rel(X,Y)〉. Then we say that G satisfies a Freiheitssatz relative to A if < X >G= A, that is the subgroup of G generated by X is isomorphic to A. In this more general language Magnus' result says that a one-relator group satisfies a Freiheitssatz relative to the free group on any proper subset of generators.

A great deal of work has gone into proving the Freiheitssatz for the class of one-relator products. These are groups of the form G = (A * B)/N(R) where R is a non-trivial,cyclically reduced word in the free product A * B of syllable length at least two. A and B are called the factors and R is called the relator. In this case the Freiheitssatz means that A and B inject into G.