A counterexample to the unit conjecture for group rings
Group Theory
2021-11-24 v4 Rings and Algebras
Abstract
The unit conjecture, commonly attributed to Kaplansky, predicts that if is a field and is a torsion-free group then the only units of the group ring are the trivial units, that is, the non-zero scalar multiples of group elements. We give a concrete counterexample to this conjecture; the group is virtually abelian and the field is order two.
Cite
@article{arxiv.2102.11818,
title = {A counterexample to the unit conjecture for group rings},
author = {Giles Gardam},
journal= {arXiv preprint arXiv:2102.11818},
year = {2021}
}
Comments
12 pages; v4 final version; v3 add corollary on group of units, reformulate proof, expand discussion; v2 add reference to conjecture in Higman's thesis