One-relator products of torsion-free groups

Abstract

If A and B are torsion-free groups, and W is a cyclically reduced word of even length in A*B, it is generally conjectured that a Freiheitssatz holds, namely that each of A and B are embedded via the natural map into the one-relator product group G = (A*B)/N(W), where N denotes normal closure. If W has length 2, then G is a free product of A and B with infinite cyclic amalgamation, and the result is obvious. The purpose of this note is to prove the Freiheitssatz in some special cases.