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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.

##### MSC:
 20F05 Generators, relations, and presentations of groups 20F06 Cancellation theory of groups; application of van Kampen diagrams 57M20 Two-dimensional complexes (manifolds) (MSC2010)
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##### References:
 [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
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