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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
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

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