zbMATH — the first resource for mathematics

On balanced presentations of the trivial group. (English) Zbl 1103.20027
Let \({\mathcal P}=\langle x_1,\dots,x_m\mid R_1,\dots,R_m\rangle\) be a balanced presentation that defines the trivial group. Magnus posed the following long-standing problem. Would it always be possible to replace the defining relator \(R_i\) (\(i=1,\dots,m\)) by a free generator of the free group \(F=F(X)\) with the set of free generators \(X=\{x_1,\dots,x_m\}\) so that the group defined by the altered presentation would still be trivial?
The author constructs a balanced presentation of the trivial group such that every such swap gives a presentation of a non-trivial group. So, the answer to the Magnus question is “no”. Some other related problems on balanced presentations of groups are discussed, too.

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: DOI
[1] Akbulut, S., Kirby, R.: A potential smooth counterexample in dimension 4 to the Poincaré conjecture, the Schoenflies conjecture, and the Andrews–Curtis conjecture. Topology 24, 375–390 (1985) · Zbl 0584.57009
[2] Andrews, J.J., Curtis, M.L.: Free groups and handlebodies. Proc. Am. Math. Soc. 16, 192–195 (1965) · Zbl 0131.38301
[3] Andrews, J.J., Curtis, M.L.: Extended Nielsen operations in free groups. Am. Math. Mon. 73, 21–28 (1966) · Zbl 0135.04403
[4] Burns, R.G., Macedonska, O.: Balanced presentations of the trivial group. Bull. Lond. Math. Soc. 25, 513–526 (1993) · Zbl 0796.20022
[5] Fenn, R., Rourke, C.: Klyachko’s methods and the solution of equations over torsion-free groups. Enseign. Math., II. Sér. 42, 49–74 (1996) · Zbl 0861.20029
[6] Gerstenhaber, M., Rothaus, O.S.: The solution of sets of equations in groups. Proc. Natl. Acad. Sci. USA 48, 1531–1533 (1962) · Zbl 0112.02504
[7] Hog-Angeloni, C., Metzler, W.: Geometric aspects of two-dimensional complexes. Lond. Math. Soc. Lect. Note Ser. 197, 1–35 (1993) · Zbl 0811.57001
[8] Hog-Angeloni, C., Metzler, W.: The Andrews–Curtis conjecture and its generalizations. Lond. Math. Soc. Lect. Note Ser. 197, 365–380 (1993) · Zbl 0814.57002
[9] Howie, J.: On pairs of 2-complexes and systems of equations over groups. J. Reine Angew. Math. 22, 475–485 (1983) · Zbl 0524.57002
[10] Howie, J.: Some remarks on a problem of J.H.C. Whitehead. Topology 324, 165–174 (1981) · Zbl 0447.20032
[11] Ivanov, S.V.: On aspherical presentations of groups. Electron. Res. Announc. Am. Math. Soc. 4, 109–114 (1998) · Zbl 0923.20024
[12] Ivanov, S.V.: The free Burnside groups of sufficiently large exponents. Int. J. Algebra Comput. 4, 1–308 (1994) · Zbl 0822.20044
[13] Klyachko, A.A.: A funny property of a sphere and equations over groups. Commun. Algebra 21, 2555–2575 (1993) · Zbl 0788.20017
[14] Kargapolov, M.I., Merzliakov, Yu.I. (eds.): Kourovka Notebook: Unsolved problems in group theory, 1st edn. Novosibirsk: Math. Institute SO AN USSR 1965
[15] Lustig, M.: Nielsen equivalence and simple-homotopy type. Proc. Lond. Math. Soc. 62, 537–562 (1991) · Zbl 0742.57003
[16] Lyndon, R.C., Schupp, P.E.: Combinatorial group theory. Springer 1977 · Zbl 0368.20023
[17] Lyndon, R.C., Schützenberger, M.P.: The equation a M =b N c P in a free group. Mich. Math. J. 9, 289–298 (1962) · Zbl 0106.02204
[18] Magnus, W., Karras, J., Solitar, D.: Combinatorial group theory. Interscience Publ. 1966 · Zbl 0138.25604
[19] Metzler, W.: Die Unterscheidung von Homotopietyp und einfachem Homotopietyp und bei zweidimensionalen Komplexen. J. Reine Angew. Math. 403, 201–219 (1990) · Zbl 0675.57002
[20] Myasnikov, A.D., Myasnikov, A.G., Shpilrain, V.: On the Andrews–Curtis equivalence. Contemp. Math. 296, 183–198 (2002) · Zbl 1010.20019
[21] Rapaport, E.S.: Groups of order 1, some properties of presentations. Acta Math. 121, 127–150 (1968) · Zbl 0159.30501
[22] Ol’shanskii, A.Yu.: Varieties in which finite groups are abelian. Mat. Sb. 126, 59–82 (1985)
[23] Ol’shanskii, A.Yu.: Geometry of defining relations in groups. Moscow: Nauka 1989; English translation: Math. Appl., Sov. Ser., vol. 70. Kluwer Acad. Publ. 1991
[24] Whitehead, J.H.C.: On adding relations to homotopy groups. Ann. Math. 42, 409–428 (1941) · Zbl 0027.26404
[25] Wright, P.: Group presentations and formal deformations. Trans. Am. 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.