On Rourke’s extension of group presentations and a cyclic version of the Andrews–Curtis conjecture
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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.References
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Additional Information
- S. V. Ivanov
- Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
- Email: ivanov@math.uiuc.edu
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 1561-1567
- MSC (2000): Primary 20F05; Secondary 57M20
- DOI: https://doi.org/10.1090/S0002-9939-05-08450-9
- MathSciNet review: 2204265