Satisfiability and group rings

• Applicant: Professor Dr. Giles Gardam
• Subject Area: Mathematics
• Term: from 2022 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 506523109

• Final Report Year: 2024

Final Report Abstract

The unit conjecture for group rings predicts that the group ring of a torsion-free group over a field has only trivial units, i.e. scalar multiples of group elements. It was known that the conjecture is false for positive characteristic fields. In this project we showed that it is also false in the remaining and arguably most important case, namely in characteristic zero. Finding integral units and counterexamples to other conjectures of Kaplansky will be the subject of future work.

Publications

• Graphs and groups with unique geodesics
  Murray Elder; Giles Gardam; Adam Piggott; Davide Spriano & Kane Townsend
  
• Non-trivial units of complex group rings
  Giles Gardam