On singular equations over torsion-free groups

Abstract

We prove a Freiheitssatz for one-relator products of torsion-free groups, where the relator has syllable length at most [Formula: see text]. This result has applications to equations over torsion-free groups: in particular a singular equation of syllable length at most [Formula: see text] over a torsion-free group has a solution in some overgroup.