Satisfiability and group rings

Group rings are key objects in many fields of mathematics including algebra, topology, operator algebras and representation theory. Fundamental questions about them remain unanswered, in particular several conjectures attributed to Kaplansky.

For torsion-free groups and field coefficients, the zero divisor conjecture predicts the absence of zero divisors and the idempotent conjecture predicts that 0 and 1 are the only idempotents in the group ring.

The direct finiteness conjecture says that left-invertible elements are invertible in group rings of arbitrary groups over fields. These conjectures have a history spanning more than 80-years.

Although special cases are known, resolving any of the conjectures in full generality seemed intractable until the recent disproof of the closely related unit conjecture.The goal of this project is to construct counterexamples to the zero divisor and direct finiteness conjectures. The latter will give the first example of a non-sofic group.

We also seek to resolve the unit conjecture in characteristic zero. Key to our approach is the application of modern solvers for Boolean satisfiability.

This paradigm shift, which was successful against the unit conjecture, shows that these problems are substantially more vulnerable to computational techniques than previously thought.

Constructing our counterexamples will require both developing our understanding of candidate groups and their properties and building a toolkit for the effective application of existing computational machinery.

The unique product property obstructs the existence of counterexamples to these conjectures and is thus of great interest.

We will answer fundamental questions about this property.Although we focus on the positive characteristic case, this project will lay serious groundwork towards the construction of counterexamples in characteristic zero to the zero divisor and idempotent conjectures and thus to the Atiyah, Baum-Connes and Farrell-Jones conjectures.