A strengthened Freiheitssatz for one-relator small cancellation free products

Abstract

Let F be a free group, freely generated by a non-empty set X and let R be a cyclically reduced word in F . Let X 0 be the subset of elements of X which occur in R or in R −1 . Let G be the quotient of F by the normal closure of R. By the classical Dehn-Magnus Freiheitssatz, if Y is a subset of X which does not contain X 0 , then the subgroup of G generated by the image of Y in G is freely generated by it.Free products of groups can be viewed as generalisations of free groups, by replacing the infinite cyclic groups generated by individual elements of X, by arbitrary groups. It is known that in special cases the corresponding version of the Freiheitssatz holds true for one-relator quotients of free products of two groups (components), for example if the relator satisfies a small cancellation condition.In this work we extend Magnus' Freiheitssatz to an arbitrary number of components and where the subset corresponding to Y above contains arbitrary number of components, provided that the relator satisfies the small cancellation condition C(8) and some further conditions. As an application we imitate, under certain conditions, the result of Magnus stating that one-relator groups have the structure of HNN extensions.The method of proof includes a combinatorial analysis of van Kampen diagrams, relying on the structure theorem for small cancellation diagrams.