In 1956-1957 Kaplansky presented a list of several questions about group rings F[G], where F is a field and G is a group. Two of those questions are concerned with whether there are nontrivial units and zero-divisors in a torsion-free group ring. In 2021, G. Gardam found the first example of nontrivial units in a group ring over a virtually abelian group and a field of order 2.
In this talk we present a computational approach we employed in search of nontrivial units and zero divisors in a torsion-free group ring. We discuss an algorithm which aims to identify product structures satisfying various combinatorial conditions arising from the topology and geometry of the units and the zero-divisors. Identifying structures with such conditions could provide examples of groups with nontrivial units and zero-divisors. This is joint work with Igor Mineyev.