TITLE: Zero divisors in group algebras of torsion-free groups of length 3
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: It is open for the group algebra over the field with $2$ elements, see 
http://arxiv.org/abs/1202.6645 and http://arxiv.org/abs/1112.1790
and references there. I saw on the website of Mikhailov an unpublished paper, where he  contributes this question to I. Rips.