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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 Ĝ, where G is a torsion-free group and Ĝ is obtained by adding a new generator t and a new relator w, is surjective only if w is conjugate to gt or gt1 where g 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
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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