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

We prove that the natural map $G \to \hat{G}$ , where $G$ is a torsion-free group and $\hat{G}$ is obtained by adding a new generator $t$ and a new relator $w$ , is surjective only if $w$ is conjugate to $g t$ 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

References

Forward citations

Publication

Received: 12 September 2000

Accepted: 6 March 2001

Published: 6 March 2001

Proposed: Cameron Gordon

Seconded: Joan Birman, Wolfgang Metzler

Authors

Marshall M Cohen
Cornell University Ithaca New York 14853-4102 USA

Colin Rourke
Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom

Volume 5, issue 1 (2001)

Marshall M Cohen and Colin Rourke