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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On Rourke's extension of group presentations and a cyclic version of the Andrews-Curtis conjecture


Author: S. V. Ivanov
Journal: Proc. Amer. Math. Soc. 134 (2006), 1561-1567
MSC (2000): Primary 20F05; Secondary 57M20
Published electronically: December 14, 2005
MathSciNet review: 2204265
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Abstract: In 1979, Rourke proposed to extend the set of cyclically reduced defining words of a group presentation $ \mathcal P$ by using operations of cyclic permutation, inversion and taking double products. He proved that iterations of these operations yield all cyclically reduced words of the normal closure of defining words of $ \mathcal P$ if the group, defined by the presentation $ \mathcal P$, is trivial. We generalize this result by proving it for every group presentation $ \mathcal P$ with an obvious exception. We also introduce a new, ``cyclic", version of the Andrews-Curtis conjecture and show that the original Andrews-Curtis conjecture with stabilizations is equivalent to its cyclic version.


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Additional Information

S. V. Ivanov
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: ivanov@math.uiuc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08450-9
PII: S 0002-9939(05)08450-9
Received by editor(s): December 28, 2004
Published electronically: December 14, 2005
Additional Notes: This research was supported in part by NSF grants DMS 00-99612 and DMS 04-00476
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.