On Conjectures of Andrews and Curtis

Ivanov, S.

doi:10.1090/proc/13710

On Conjectures of Andrews and Curtis
HTML articles powered by AMS MathViewer

by S. V. Ivanov PDF

Proc. Amer. Math. Soc. 146 (2018), 2283-2298 Request permission

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

J. J. Andrews and M. L. Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965), 192–195. MR 173241, DOI 10.1090/S0002-9939-1965-0173241-8
J. J. Andrews and M. L. Curtis, Extended Nielsen operations in free groups, Amer. Math. Monthly 73 (1966), 21–28. MR 195928, DOI 10.2307/2313917
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, DOI 10.1007/3-7643-7447-0_{2}
R. G. Burns and Olga Macedońska, Balanced presentations of the trivial group, Bull. London Math. Soc. 25 (1993), no. 6, 513–526. MR 1245076, DOI 10.1112/blms/25.6.513
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, DOI 10.1142/S0218196703001365
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, DOI 10.1017/CBO9780511629358.014
Sergei V. Ivanov, The free Burnside groups of sufficiently large exponents, Internat. J. Algebra Comput. 4 (1994), no. 1-2, ii+308. MR 1283947, DOI 10.1142/S0218196794000026
S. V. Ivanov, Recognizing the 3-sphere, Illinois J. Math. 45 (2001), no. 4, 1073–1117. MR 1894888
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, DOI 10.1090/S0002-9939-05-08450-9
S. V. Ivanov, On balanced presentations of the trivial group, Invent. Math. 165 (2006), no. 3, 525–549. MR 2242626, DOI 10.1007/s00222-005-0497-1
S. V. Ivanov, The computational complexity of basic decision problems in 3-dimensional topology, Geom. Dedicata 131 (2008), 1–26. MR 2369189, DOI 10.1007/s10711-007-9210-4
Martin Lustig, Nielsen equivalence and simple-homotopy type, Proc. London Math. Soc. (3) 62 (1991), no. 3, 537–562. MR 1095232, DOI 10.1112/plms/s3-62.3.537
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
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
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, DOI 10.1070/SM1995v186n05ABEH000037
A. G. Myasnikov, Extended Nielsen transformations and the trivial group, Mat. Zametki 35 (1984), no. 4, 491–495 (Russian). MR 744511
Alexei D. Miasnikov, Genetic algorithms and the Andrews-Curtis conjecture, Internat. J. Algebra Comput. 9 (1999), no. 6, 671–686. MR 1727164, DOI 10.1142/S0218196799000370
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, DOI 10.1090/conm/296/05074
A. Yu. Ol′shanskiĭ, Geometry of defining relations in groups, Mathematics and its Applications (Soviet Series), vol. 70, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the 1989 Russian original by Yu. A. Bakhturin. MR 1191619, DOI 10.1007/978-94-011-3618-1
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv.org, November 11, 2002.
G. Perelman, Ricci flow with surgery on three-manifolds, arXiv.org, March 10, 2003.
G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv.org, July 17, 2003.
Elvira Strasser Rapaport, Remarks on groups of order $1$, Amer. Math. Monthly 75 (1968), 714–720. MR 236251, DOI 10.2307/2315181
Elvira Strasser Rapaport, Groups of order $1$: Some properties of presentations, Acta Math. 121 (1968), 127–150. MR 229704, DOI 10.1007/BF02391911
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
Abigail Thompson, Thin position and the recognition problem for $S^3$, Math. Res. Lett. 1 (1994), no. 5, 613–630. MR 1295555, DOI 10.4310/MRL.1994.v1.n5.a9
Perrin Wright, Group presentations and formal deformations, Trans. Amer. Math. Soc. 208 (1975), 161–169. MR 380813, DOI 10.1090/S0002-9947-1975-0380813-5