Satisfiability and group rings

Project Information

SATURN

Grant agreement ID: 101076148

DOI

10.3030/101076148

EC signature date 16 July 2023

Start date 1 October 2023

End date 30 September 2028

Funded under

• European Research Council (ERC)

Total cost

€ 1 207 875,00

EU contribution

€ 1 207 875,00

Coordinated by

RHEINISCHE FRIEDRICH-WILHELMS-UNIVERSITAT BONN
Germany

Periodic Reporting for period 1 - SATURN (Satisfiability and group rings)

Reporting period: 2023-10-01 to 2026-03-31

Summary of the context and overall objectives of the project

This project is concerned with algebraic objects called group rings. They are important in many different mathematical subjects, such as algebraic topology, representation theory, and operator algebras. There is a series of fundamental conjectures about group rings attributed to Kaplansky which claim to characterize certain elements in group rings, such as their units. They can be thought of as conjectures of tameness. The aim of this project is to produce counterexamples to these conjectures, indicating that group rings contains surprisingly wild elements. As the Kaplansky conjectures connect to other questions about groups, these counterexamples could lead to resolving other major open problems.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

We have both investigated which groups are potential counterexamples and attempted to advance our techniques for constructing counterexamples in a given candidate group. So far the main achievement has been producing the first example of a non-trivial unit in the group ring of a torsion-free group over a field of characteristic zero, which was one of the three main objectives of this project.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The main achievement, namely constructing non-trivial units in characteristic zero, opens up the possibility that zero divisors could also exist in characteristic zero, which would disprove the so-called Atiyah conjecture.