A counterexample to the unit conjecture for group rings

Abstract

The unit conjecture, commonly attributed to Kaplansky, predicts that if $K$ is a field and $G$ is a torsion-free group then the only units of the group ring $K[G]$ 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.

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

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