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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: [email protected]
MICHAEL TWEEDALE
Affiliation:
Department of Mathematics, University of Bristol, Bristol BS8 1TW. [email protected]
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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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References

REFERENCES

[1]Bestvina, M. and Brady, N.. Morse theory and finiteness properties of groups. Invent. Math. 129 (1997), 445–470.Google Scholar
[2]Bridson, M. R. and Tweedale, M.. Constructing presentations of finite-index subgroups of right-angled Artin groups. In preparationGoogle Scholar
[3]Epstein, D. B. A.. Finite presentations of groups and 3-manifolds. Quart. J. Math. Oxford Ser. (2) 12 (1961), 205–212.Google Scholar
[4]Gruenberg, K. and Linnell, P.. Generation gaps and abelianized defects of free products. J. Group Theory, to appear.Google Scholar
[5]Harlander, J.. Some Aspects of Efficiency. Groups–Korea '98 (Pusan) (de Gruyter, 2000), pp. 165–180.Google Scholar
[6]Hausmann, J.-C. and Weinberger, S.. Caractéristiques d'Euler et groupes fondamentaux des variétés de dimension 4, Comment. Math. Helv. 60 (1985), 139–144.Google Scholar
[7]Hillman, J. A.. An homology 4-sphere group with negative deficiency. Enseign. Math. (2) 48 (2002), 259–262.Google Scholar
[8]Hog, C., Lustig, M. and Metzler, W.. Presentation classes, 3-manifolds and free products. Geometry and topology (College Park, Md., 1983/84) Lecture Notes in Math. vol. 1167, (Springer, 1985), pp. 154–167.Google Scholar
[9]Howie, J.. On locally indicable groups. Math. Z. 180 (1982), 445–461.Google Scholar
[10]Howie, J.. Bestvina–Brady groups and the plus construction. Math. Proc. Camb. Phil. Soc. 127 (1999), 487–493.Google Scholar
[11]Johnson, F. E. A.. Stable modules and the D(2)-problem. London Math. Soc. Lecture Note Ser. vol. 301, (Cambridge University Press, 2003).Google Scholar
[12]Lustig, M.. On the rank, the deficiency and the homological dimension of groups: the computation of a lower bound via Fox ideals. Topology and Combinatorial Group Theory (Hanover, NH, 1986/1987; Enfield, NH, 1988). Lecture Notes in Math. vol. 1440, (Springer, 1990), pp. 164–174.Google Scholar
[13]Lustig, M.. Non-efficient torsion-free groups exist. Comm. Algebra 23 (1995), 215–218.CrossRefGoogle Scholar
[14]Swan, R. G.. Minimal resolutions for finite groups. Topology 4 (1965), 193–208.Google Scholar
[15]Wall, C. T. C.. Finiteness conditions for CW-complexes. Ann. of Math. (2) 81 (1965), 56–69.Google Scholar