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 | References | Similar Articles | Additional Information

Abstract: In 1979, Rourke proposed to extend the set of cyclically reduced defining words of a group presentation 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 if the group, defined by the presentation , is trivial. We generalize this result by proving it for every group presentation 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.

**[1]**J. J. Andrews and M. L. Curtis,*Free groups and handlebodies*, Proc. Amer. Math. Soc.**16**(1965), 192–195. MR**0173241**, 10.1090/S0002-9939-1965-0173241-8**[2]**J. J. Andrews and M. L. Curtis,*Extended Nielsen operations in free groups*, Amer. Math. Monthly**73**(1966), 21–28. MR**0195928****[3]**R. G. Burns and Olga Macedońska,*Balanced presentations of the trivial group*, Bull. London Math. Soc.**25**(1993), no. 6, 513–526. MR**1245076**, 10.1112/blms/25.6.513**[4]**Sergei V. Ivanov,*The free Burnside groups of sufficiently large exponents*, Internat. J. Algebra Comput.**4**(1994), no. 1-2, ii+308. MR**1283947**, 10.1142/S0218196794000026**[5]**Alexei D. Myasnikov, Alexei G. Myasnikov, and Vladimir Shpilrain,*On the Andrews-Curtis equivalence*, Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), Contemp. Math., vol. 296, Amer. Math. Soc., Providence, RI, 2002, pp. 183–198. MR**1921712**, 10.1090/conm/296/05074**[6]**A. Yu. Ol′shanskiĭ,*Geometriya opredelyayushchikh sootnoshenii v gruppakh*, \cyr Sovremennaya Algebra. [Modern Algebra], “Nauka”, Moscow, 1989 (Russian). With an English summary. MR**1024791****[7]**C. P. Rourke,*Presentations and the trivial group*, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977) Lecture Notes in Math., vol. 722, Springer, Berlin, 1979, pp. 134–143. MR**547460****[8]**Fabio Scarabotti,*On the presentations of the trivial group*, J. Group Theory**2**(1999), no. 3, 329–333. MR**1696318**, 10.1515/jgth.1999.022

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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

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.