It is easy to see that there are no group divisors of length 2 in a group algebra of a torsionfree group. I saw somewhere mentioned that it is possible to do it for length 3. How?
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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 I saw on the website of Mikhailov an unpublished paper, where he contributes this question to I. Rips. 

