Abstract by Lindsay Soelberg
Group Algebras of Torsion-free Groups: Zero Divisors or not?
Given a group G and a field F let F[G] denote the group algebra. There is an old conjecture from Kaplansky dating back to 1956 that says if G is torsion-free then F[G] has no zero divisors. It is easy to show that this conjecture is true when G is a unique product group. It is possible to exhaustively search for torsion-free groups without the unique product property on sets of limited size. Searching for groups with no unique products is a first step in finding a counterexample to this conjecture. This comprehensive inquiry has shown that if A*B has no unique product where A,B are subsets of G, then |A| + |B| > 12. Further, there exist sets A,B that are subsets of G with |A| = |B| = 8 which has no unique product.