The surjectivity problem for one-generator,, one-relator extensions of torsion-free groups

We use Klyachko's methods to prove that the natural map G to G-hat, where G is a torsion-free group and G-hat is obtained by adding a new generator t and a new relator w, is surjective only if w is conjugate to gt or gt^{-1} for some g in G. This solves a special case of the surjectivity problem for group extensions, raised by Cohen [Whitehead torsion, group extensions, and Zeeman's conjecture in high dimensions, Topology, 16 (1977) 79--88].