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Deficiency and abelianized deficiency of some virtually free groups

Published online by Cambridge University Press:  01 September 2007

MARTIN R. BRIDSON  and
MICHAEL TWEEDALE
MARTIN R. BRIDSON
Affiliation:
Department of Mathematics, Imperial College, London SW7 2AZ. email: m.bridson@imperial.ac.uk
MICHAEL TWEEDALE
Affiliation:
Department of Mathematics, University of Bristol, Bristol BS8 1TW. emailm.tweedale@bristol.ac.uk
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Abstract

Let Qm be the HNN extension of Z/m × Z/m where the stable letter conjugates the first factor to the second. We explore small presentations of the groups Γm,n = Qm* Qn. We show that for certain choices of (m,n), for example (2,3), the group Γm,n has a relation gap unless it admits a presentation with at most 3 defining relations, and we establish restrictions on the possible form of such a presentation. We then associate to each (m,n) a 3-complex with 16 cells. This 3-complex is a counterexample to the D(2) conjecture if Γm,n has a relation gap.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 2 , September 2007 , pp. 257 - 264
Copyright
Copyright © Cambridge Philosophical Society 2007

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