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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.

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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