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I was told that there is a theorem stating that if $G$ is a finitely presented nontrivial linear group then it is not possible to take a presentation for $G$ and make it a presentation for the trivial group by adding an equal number of generators and relations.

Does anyone know a reference for this theorem or how to prove it?

Thanks!

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    $\begingroup$ You should ask on MO $\endgroup$ – YCor Mar 2 '18 at 23:55
  • $\begingroup$ I also suggest you ask your source for the reference to this claim. All I can say is that the problem reduces to the case when $G$ is a finite group. $\endgroup$ – Moishe Kohan Mar 5 '18 at 17:17
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I think you are referring to the following paper:

Gerstenhaber, Murray; Rothaus, Oscar S. The solution of sets of equations in groups. Proc. Nat. Acad. Sci. U.S.A. 48 1962 1531–1533.

Also of interest (and more easily available, for instance on the arXiv) is the following:

Klyachko, A. A. The Kervaire-Laudenbach conjecture and presentations of simple groups. (Russian. Russian summary) Algebra Logika 44 (2005), no. 4, 399--437, 512; translation in Algebra Logic 44 (2005), no. 4, 219–242

The term you want to search for is the "Kervaire-Laudenbach conjecture".

Edit: I found these slides by Andreas Thom, which include a sketch of the proof of the Gerstenhaber-Rothaus theorem and an indication as to how solving equations over groups is related to the possibility of making a non-trivial group a trivial one by adding one generator and one relation.

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  • $\begingroup$ Very nice: The paper of Gerstenhaber and Rothaus contains a proof of the theorem that OP refers to, in the case when $G$ is finite and some nonzero degree condition is satisfied by the relators added to the group. The finiteness condition can be easily eliminated (when $G$ is residually finite e.g. linear) but I am not so sure about the degree condition. $\endgroup$ – Moishe Kohan Mar 6 '18 at 20:25

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