Freiheitssatz for amalgamated products of free groups over maximal cyclic subgroups

In 1930, Wilhelm Magnus introduced the so-called Freiheitssatz: Let F be a free group with basis X and let r be a cyclically reduced element of F which contains a basis element x E X , then every non -trivial element of the normal closure of r in F contains the basis element x . Equivalently, the subgroup freely generated by X \{ x } embeds canonically into the quotient group F /x x r y y F . In this article, we want to introduce a Freiheitssatz for amalgamated products G = A * U B of free groups A and B , where U is a maximal cyclic subgroup in A and B : If an element r of G is neither conjugate to an element of A nor B , then the factors A , B embed canonically into G /x x r y y G . (c) 2024 Published by Elsevier Inc.