2019-04-16 01:09:47 8 Comments

Is it decidable whether two given elements of ${\rm GL}(n,{\mathbb Z})$ generate a free group of rank 2?

This is a simple question that I have been asking people for the past couple of years, but nobody has known the answer, so I thought I would try here.

The Tits alternative is known to be (effectively) decidable for finitely generated subgroups of ${\rm GL}(n,{\mathbb Z})$, but that is not helpful here.

*Added*: From what Misha says, the answer to the general problem might be unknown, but it is likely to be undecidable. An easier question would be, assuming that the group in question is not virtually solvable, can we find a nonabelian free subgroup (with proof). I think the answer to that might be yes, using pingpong.

This second question is answered positively here

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