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Towers, ladders and the B. B. Newman Spelling Theorem

Part of: Special aspects of infinite or finite groups Low-dimensional topology

Published online by Cambridge University Press:  09 April 2009

G. Christopher Hruska  and
Daniel T. Wise
G. Christopher Hruska
Affiliation:
Department of Mathematics Cornell UniversityIthaca, NY 14853USA e-mail: chruska@math.cornell.edu
Daniel T. Wise
Affiliation:
Department of Mathematics and Statistics McGill UniversityMontreal, Quebec H3A ZK6Canada e-mail: wise@math.mcgill.ca
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Abstract

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The Spelling Theorem of B. B. Newman states that for a one-relator group (a1, … | Wn), any nontrivial word which represents the identity must contain a (cyclic) subword of W±n longer than Wn−1. We provide a new proof of the Spelling Theorem using towers of 2-complexes. We also give a geometric classification of reduced disc diagrams in one-relator groups with torsion. Either the disc diagram has three 2-cells which lie almost entuirly along the bounday, or the disc diagram looks like a ladder. We use this ladder theorem to prove that a large class of one-relator groups with torsion are locally quasiconvex.

Keywords

One-relator groupstowersstaggered 2-complexeslocal quasiconvexity

MSC classification

Secondary: 20F06: Cancellation theory; application of van Kampen diagrams 57M07: Topological methods in group theory 20F67: Hyperbolic groups and nonpositively curved groups

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 71 , Issue 1 , August 2001 , pp. 53 - 69
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Bogley, W. A. and Pride, S. J., ‘Calculating generators of π2’, in: Two-dimensional homotopy and combinatorial group theory (eds. Hog-Angeloni, C.. Metzler, W. and Sieradski, A. J.) (Cambridge Univ. Press, Cambridge, 1993) pp. 157–188.CrossRefGoogle Scholar
[2]Howie, J., ‘On pairs of 2-complexes and systems of equations over groups’, J. Reine Angew. Math. 324 (1981), 165–174.Google Scholar
[3]Howie, J., ‘How to generalize one-relator group theory’. in: Combinatorial group theory and topology (eds. Gersten, S. M. and Stallings, J. R.) (Princeton Univ. Press, Princeton, N.J., 1987) pp. 53–78.Google Scholar
[4]Howie, J. and Pride, S. J., ‘A spelling theorem for staggered generalized 2-complexes, with applications’, Invent. Math. 76 (1984), 55–74.CrossRefGoogle Scholar
[5]Lyndon, R. C., ‘Cohomology theory of groups with a single defining relation’, Ann. of Math. (2) 52 (1950), 650–665.CrossRefGoogle Scholar
[6]Lyndon, R. C. and Schupp, P. E., Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89 (Springer, Berlin, 1977).Google Scholar
[7]Magnus, W., ‘Über diskontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz)’, J. Reine Angew. Math. 163 (1930), 141–165.Google Scholar
[8]Magnus, W., ‘Das Identitätsproblem für Gruppen mit einer definererden Relation’, Math. Ann. 106 (1932), 295–307.CrossRefGoogle Scholar
[9]McCammond, J. P. and Wise, D. T., ‘Coherence, local quasiconvexity and the perimeter of 2- complexes’, preprint, 1999.Google Scholar
[10]McCammond, J. P., ‘Fans and ladders in small cancellation theory’, Proc. London Math. Soc., to appear.Google Scholar
[11]Newman, B. B., ‘Some results on one-relator groups’, Bull. Amer. Math. Soc. 74 (1968), 568–571.CrossRefGoogle Scholar
[12]Papakyriakopoulos, C. D., ‘On Dehn's lemma and the asphericity of knots’, Ann. of Math. (2) 66 (1957), 1–26.CrossRefGoogle Scholar
[13]Short, H., ‘Quasiconvexity and a theorem of Howson's’, in: Group theory from a geometrical viewpoint (Trieste, 1990) (eds. Ghys, É., Haefliger, A. and Verjovsky, A.) (World Sci. Publishing, River Edge, NJ, 1991) pp. 168–176.Google Scholar