By Derek Holt

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


Related Questions

Sponsored Content

7 Answered Questions

[SOLVED] Which group does not satisfy the Tits alternative?

1 Answered Questions

[SOLVED] For which groups is (non-)left orderability decidable?

3 Answered Questions

[SOLVED] Can a group be a universal Turing machine?

0 Answered Questions

0 Answered Questions

2 Answered Questions

[SOLVED] Hilbert's 10th problem and nilpotent groups

4 Answered Questions

[SOLVED] Trees in groups of exponential growth

2 Answered Questions

[SOLVED] The generalized word problem vs. the uniform generalized word problem

  • 2011-08-05 17:23:44
  • Benjamin Steinberg
  • 1047 View
  • 12 Score
  • 2 Answer
  • Tags:

2 Answered Questions

[SOLVED] Free subgroups vs law

Sponsored Content