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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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