Geometry & Topology, Vol. 5 (2001) Paper no. 5, pages 127--142.

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

Marshall M Cohen, Colin Rourke

Abstract. 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].

Keywords. Surjectivity problem, torsion-free groups, Whitehead torsion, Kervaire conjecture

AMS subject classification. Primary: 20E22, 20F05. Secondary: 57M20, 57Q10.

DOI: 10.2140/gt.2001.5.127

E-print: arXiv:math.GR/0009101

Submitted to GT on 12 September 2000. Paper accepted 6 March 2001. Paper published 6 March 2001.

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Marshall M Cohen, Colin Rourke
Cornell University, Ithaca NY, 14853-4102, USA
Warwick University, Coventry, CV4 7AL, UK
Email: marshall@math.cornell.edu, cpr@maths.warwick.ac.uk

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