Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On Conjectures of Andrews and Curtis


Author: S. V. Ivanov
Journal: Proc. Amer. Math. Soc. 146 (2018), 2283-2298
MSC (2010): Primary 20F05, 20F06, 57M20
DOI: https://doi.org/10.1090/proc/13710
Published electronically: March 9, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the original Andrews-Curtis conjecture on balanced presentations of the trivial group is equivalent to its ``cyclic'' version in which, in place of arbitrary conjugations, one can use only cyclic permutations. This, in particular, proves a satellite conjecture of Andrews and Curtis [Amer. Math. Monthly 73 (1966), 21-28]. We also consider a more restrictive ``cancellative'' version of the cyclic Andrews-Curtis conjecture with and without stabilizations and show that the restriction does not change the Andrews-Curtis conjecture when stabilizations are allowed. On the other hand, the restriction makes the conjecture false when stabilizations are not allowed.


References [Enhancements On Off] (What's this?)

  • [1] J. J. Andrews and M. L. Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965), 192-195. MR 0173241, https://doi.org/10.2307/2033843
  • [2] J. J. Andrews and M. L. Curtis, Extended Nielsen operations in free groups, Amer. Math. Monthly 73 (1966), 21-28. MR 0195928, https://doi.org/10.2307/2313917
  • [3] Alexandre V. Borovik, Alexander Lubotzky, and Alexei G. Myasnikov, The finitary Andrews-Curtis conjecture, Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math., vol. 248, Birkhäuser, Basel, 2005, pp. 15-30. MR 2195451, https://doi.org/10.1007/3-7643-7447-0_2
  • [4] R. G. Burns and Olga Macedońska, Balanced presentations of the trivial group, Bull. London Math. Soc. 25 (1993), no. 6, 513-526. MR 1245076, https://doi.org/10.1112/blms/25.6.513
  • [5] George Havas and Colin Ramsay, Breadth-first search and the Andrews-Curtis conjecture, Internat. J. Algebra Comput. 13 (2003), no. 1, 61-68. MR 1970867, https://doi.org/10.1142/S0218196703001365
  • [6] Cynthia Hog-Angeloni and Wolfgang Metzler, The Andrews-Curtis conjecture and its generalizations, Two-dimensional homotopy and combinatorial group theory, London Math. Soc. Lecture Note Ser., vol. 197, Cambridge Univ. Press, Cambridge, 1993, pp. 365-380. MR 1279186, https://doi.org/10.1017/CBO9780511629358.014
  • [7] Sergei V. Ivanov, The free Burnside groups of sufficiently large exponents, Internat. J. Algebra Comput. 4 (1994), no. 1-2, ii+308. MR 1283947, https://doi.org/10.1142/S0218196794000026
  • [8] S. V. Ivanov, Recognizing the 3-sphere, Illinois J. Math. 45 (2001), no. 4, 1073-1117. MR 1894888
  • [9] S. V. Ivanov, On Rourke's extension of group presentations and a cyclic version of the Andrews-Curtis conjecture, Proc. Amer. Math. Soc. 134 (2006), no. 6, 1561-1567. MR 2204265, https://doi.org/10.1090/S0002-9939-05-08450-9
  • [10] S. V. Ivanov, On balanced presentations of the trivial group, Invent. Math. 165 (2006), no. 3, 525-549. MR 2242626, https://doi.org/10.1007/s00222-005-0497-1
  • [11] S. V. Ivanov, The computational complexity of basic decision problems in 3-dimensional topology, Geom. Dedicata 131 (2008), 1-26. MR 2369189, https://doi.org/10.1007/s10711-007-9210-4
  • [12] Martin Lustig, Nielsen equivalence and simple-homotopy type, Proc. London Math. Soc. (3) 62 (1991), no. 3, 537-562. MR 1095232, https://doi.org/10.1112/plms/s3-62.3.537
  • [13] Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
  • [14] Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1966. MR 0207802
  • [15] S. V. Matveev, Algorithms for the recognition of the three-dimensional sphere (after A. Thompson), Mat. Sb. 186 (1995), no. 5, 69-84 (Russian, with Russian summary); English transl., Sb. Math. 186 (1995), no. 5, 695-710. MR 1341085, https://doi.org/10.1070/SM1995v186n05ABEH000037
  • [16] A. G. Myasnikov, Extended Nielsen transformations and the trivial group, Mat. Zametki 35 (1984), no. 4, 491-495 (Russian). MR 744511
  • [17] Alexei D. Miasnikov, Genetic algorithms and the Andrews-Curtis conjecture, Internat. J. Algebra Comput. 9 (1999), no. 6, 671-686. MR 1727164, https://doi.org/10.1142/S0218196799000370
  • [18] 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, https://doi.org/10.1090/conm/296/05074
  • [19] A. Yu. Ol$ \prime$shanskiĭ, Geometry of defining relations in groups, translated from the 1989 Russian original by Yu.A. Bakhturin, Mathematics and its Applications (Soviet Series), vol. 70, Kluwer Academic Publishers Group, Dordrecht, 1991. MR 1191619
  • [20] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv.org, November 11, 2002.
  • [21] G. Perelman, Ricci flow with surgery on three-manifolds, arXiv.org, March 10, 2003.
  • [22] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv.org, July 17, 2003.
  • [23] Elvira Strasser Rapaport, Remarks on groups of order $ 1$, Amer. Math. Monthly 75 (1968), 714-720. MR 0236251, https://doi.org/10.2307/2315181
  • [24] Elvira Strasser Rapaport, Groups of order $ 1$: Some properties of presentations, Acta Math. 121 (1968), 127-150. MR 0229704, https://doi.org/10.1007/BF02391911
  • [25] Joachim H. Rubinstein, An algorithm to recognize the $ 3$-sphere, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 601-611. MR 1403961
  • [26] Abigail Thompson, Thin position and the recognition problem for $ S^3$, Math. Res. Lett. 1 (1994), no. 5, 613-630. MR 1295555, https://doi.org/10.4310/MRL.1994.v1.n5.a9
  • [27] Perrin Wright, Group presentations and formal deformations, Trans. Amer. Math. Soc. 208 (1975), 161-169. MR 0380813, https://doi.org/10.2307/1997282

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 20F05, 20F06, 57M20

Retrieve articles in all journals with MSC (2010): 20F05, 20F06, 57M20


Additional Information

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

DOI: https://doi.org/10.1090/proc/13710
Received by editor(s): August 30, 2015
Received by editor(s) in revised form: June 22, 2016
Published electronically: March 9, 2018
Additional Notes: The author was supported in part by the National Science Foundation, grant DMS 09-01782
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society