#### Volume 5, issue 1 (2001)

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

### Marshall M Cohen and Colin Rourke

Geometry & Topology 5 (2001) 127–142
 arXiv: math.GR/0009101
##### Abstract

We prove that the natural map $G\to \stackrel{̂}{G}$, where $G$ is a torsion-free group and $\stackrel{̂}{G}$ is obtained by adding a new generator $t$ and a new relator $w$, is surjective only if $w$ is conjugate to $gt$ or $g{t}^{-1}$ where $g\in G$. This solves a special case of the surjectivity problem for group extensions, raised by Cohen [Topology 16 (1977) 79–88].

##### Keywords
surjectivity problem, torsion-free groups, Whitehead torsion, Kervaire conjecture
##### Mathematical Subject Classification 2000
Primary: 20E22, 20F05
Secondary: 57M20, 57Q10