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This will be an ambiguous question.

I am interested in various examples that appear in the literature of verbal subgroups of free groups, but which are not part of the "classical examples" like derived subgroups, lower central series, Burnside's verbal subgroup or standard ways of combining the above.

The crazier and harder to prove they are verbal, the better. Also, they don't need to be defined in a universal way for all free groups. If you can define them only for the infinitely generated countable free group I would still be happy.

If we have two countable groups A and B, then $[A,B]\subset A*B$ is a free group. Can the structure of A and B give rise to some exotic verbal subgroup of $[A,B]$? Can we impose conditions on A and B that will give rise to these verbal subgroups? (I was thinking at A,B-perfect groups, or reduced free groups).

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    $\begingroup$ What is the motivation for searching suvgroups whose verbality is crazy to prove? $\endgroup$
    – Ilya Bogdanov
    Commented Oct 22, 2015 at 17:30
  • $\begingroup$ I want to see in what conditions the non-abelian tensor product of a group G, acting by conjugation on itself, $G\otimes G$ is a reduced free group. If not him, at least something similar to him. He is the quotient of $[G,\overline{G}]\subset G*\overline{G}$ where $\overline{G}$ is just a copy of $G$. So all I need to do is prove is that the kernel of this projection is verbal. But I doubt it is true in general. $\endgroup$
    – Alin Galatan
    Commented Oct 22, 2015 at 17:42
  • $\begingroup$ A paper where you can see who the Kernel is: download.springer.com/static/pdf/512/…*~hmac=178bfde5b6b4481665e0b3ecf50d793193a203b2c749fca0ea6247a1e09c54e2 $\endgroup$
    – Alin Galatan
    Commented Oct 22, 2015 at 17:43

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