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.

arXiv comment12 pages; v4 final version; v3 add corollary on group of units, reformulate proof, expand discussion; v2 add reference to conjecture in Higman's thesis

References in corpus (5)

• The Baum-Connes conjecture for hyperbolic groups
• The Farrell-Jones conjecture for arbitrary lattices in virtually connected Lie groups
• Rips-Segev torsion-free groups without the unique product property
• Rips construction without unique product
• Finite index subgroups without unique product in graphical small cancellation groups

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• A note on morphisms to wreath products
• Nonexistence of exact Lagrangian tori in affine conic bundles over ${\mathbb{C}}^n$
• Non-Abelian extensions of groupoids and their groupoid rings
• More Counterexamples to the Unit Conjecture for Group Rings
• On the counterexamples to the unit conjecture for group rings