# zbMATH — the first resource for mathematics

On conjectures of Andrews and Curtis. (English) Zbl 06852821
Summary: 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 J. J. Andrews and M. L. Curtis [Am. Math. Mon. 73, 21–28 (1966; Zbl 0135.04403)]. 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.
##### MSC:
 20F05 Generators, relations, and presentations of groups 20F06 Cancellation theory of groups; application of van Kampen diagrams 57M20 Two-dimensional complexes (manifolds) (MSC2010)
Full Text:
##### References:
 [1] Andrews, J. J.; Curtis, M. L., Free groups and handlebodies, Proc. Amer. Math. Soc., 16, 192-195 (1965) · Zbl 0131.38301 [2] Andrews, J. J.; Curtis, M. L., Extended Nielsen operations in free groups, Amer. Math. Monthly, 73, 21-28 (1966) · Zbl 0135.04403 [3] Borovik, Alexandre V.; Lubotzky, Alexander; Myasnikov, Alexei G., The finitary Andrews-Curtis conjecture. Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math. 248, 15-30 (2005), Birkh\"auser, Basel · Zbl 1114.20011 [4] Burns, R. G.; Macedo\'nska, Olga, Balanced presentations of the trivial group, Bull. London Math. Soc., 25, 6, 513-526 (1993) · Zbl 0796.20022 [5] Havas, George; Ramsay, Colin, Breadth-first search and the Andrews-Curtis conjecture, Internat. J. Algebra Comput., 13, 1, 61-68 (2003) · Zbl 1059.20029 [6] Hog-Angeloni, Cynthia; Metzler, Wolfgang, The Andrews-Curtis conjecture and its generalizations. Two-dimensional homotopy and combinatorial group theory, London Math. Soc. Lecture Note Ser. 197, 365-380 (1993), Cambridge Univ. Press, Cambridge · Zbl 0814.57002 [7] Ivanov, Sergei V., The free Burnside groups of sufficiently large exponents, Internat. J. Algebra Comput., 4, 1-2, ii+308 pp. (1994) · Zbl 0822.20044 [8] Ivanov, S. V., Recognizing the 3-sphere, Illinois J. Math., 45, 4, 1073-1117 (2001) · Zbl 1002.57037 [9] Ivanov, S. V., On Rourke’s extension of group presentations and a cyclic version of the Andrews-Curtis conjecture, Proc. Amer. Math. Soc., 134, 6, 1561-1567 (2006) · Zbl 1101.20020 [10] Ivanov, S. V., On balanced presentations of the trivial group, Invent. Math., 165, 3, 525-549 (2006) · Zbl 1103.20027 [11] Ivanov, S. V., The computational complexity of basic decision problems in 3-dimensional topology, Geom. Dedicata, 131, 1-26 (2008) · Zbl 1146.57025 [12] Lustig, Martin, Nielsen equivalence and simple-homotopy type, Proc. London Math. Soc. (3), 62, 3, 537-562 (1991) · Zbl 0742.57003 [13] Lyndon, Roger C.; Schupp, Paul E., Combinatorial group theory, xiv+339 pp. (1977), Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York · Zbl 0997.20037 [14] Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald, Combinatorial group theory: Presentations of groups in terms of generators and relations, xii+444 pp. (1966), Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney · Zbl 1130.20307 [15] Matveev, S. V., Algorithms for the recognition of the three-dimensional sphere (after A. Thompson), Mat. Sb.. Sb. Math., 186 186, 5, 695-710 (1995) · Zbl 0849.57010 [16] Myasnikov, A. G., Extended Nielsen transformations and the trivial group, Mat. Zametki, 35, 4, 491-495 (1984) [17] Miasnikov, Alexei D., Genetic algorithms and the Andrews-Curtis conjecture, Internat. J. Algebra Comput., 9, 6, 671-686 (1999) · Zbl 0949.20022 [18] Myasnikov, Alexei D.; Myasnikov, Alexei G.; Shpilrain, Vladimir, On the Andrews-Curtis equivalence. Combinatorial and geometric group theory, New York, 2000/Hoboken, NJ, 2001, Contemp. Math. 296, 183-198 (2002), Amer. Math. Soc., Providence, RI · Zbl 1010.20019 [19] Ol $$\prime$$ shanski\u\i, A. Yu., Geometry of defining relations in groups, translated from the 1989 Russian original by Yu. A. Bakhturin, Mathematics and its Applications (Soviet Series) 70, xxvi+505 pp. (1991), Kluwer Academic Publishers Group, Dordrecht · Zbl 0732.20019 [20] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv.org, November 11, 2002. · Zbl 1130.53001 [21] G. Perelman, Ricci flow with surgery on three-manifolds, arXiv.org, March 10, 2003. · Zbl 1130.53002 [22] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv.org, July 17, 2003. · Zbl 1130.53003 [23] Rapaport, Elvira Strasser, Remarks on groups of order $$1$$, Amer. Math. Monthly, 75, 714-720 (1968) · Zbl 0175.29601 [24] Rapaport, Elvira Strasser, Groups of order $$1$$: Some properties of presentations, Acta Math., 121, 127-150 (1968) · Zbl 0159.30501 [25] Rubinstein, Joachim H., An algorithm to recognize the $$3$$-sphere. Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Z\`“urich, 1994, 601-611 (1995), Birkh\'”auser, Basel · Zbl 0864.57009 [26] Thompson, Abigail, Thin position and the recognition problem for $$S^3$$, Math. Res. Lett., 1, 5, 613-630 (1994) · Zbl 0849.57009 [27] Wright, Perrin, Group presentations and formal deformations, Trans. Amer. Math. Soc., 208, 161-169 (1975) · Zbl 0318.57010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.