It is easy to see that there are no group divisors of length 2 in a group algebra of a torsion-free group. I saw somewhere mentioned that it is possible to do it for length 3. How?
It is open for the group algebra over the field with $2$ elements, see
and references there. I saw on the website of Mikhailov an unpublished paper, where he contributes this question to I. Rips.