Kate Juschenko: Finitely presented groups related to Kaplansky's direct finiteness conjecture

Abstract: We study the following conjecture of Kaplansky. For every discrete group G and every field K, the group ring K[G] is directly finite, i.e., for every a,b in K[G] the equation ab=1 implies ba=1. We describe finitely presented groups that are universal for existence of one--sided invertible elements in a group algebra. To test Kaplansky's conjecture it will be enough to test it on these universal groups. We show that if rank(a)<= 7 and rank(b)<=5 then the equation ab=1 implies ba=1 over the field with two elements. This is joint work with Ken Dykema.