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arXiv · 2603.22640

On the trivial units property and the unique product property

Heiko Dietrich·Melissa Lee·Andre Nies·Marc Vinyals
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Abstract

We report on some computational experiments related to the trivial units property and unique product property for group rings of torsion-free groups. These properties are related to Kaplansky's unit and zero-divisor conjectures. Our investigations include a classification of certain symmetric non-trivial units in the binary group ring of the Hantzsche-Wendt group; this group was used in Gardam's refutal of Kaplansky's unit conjecture. We also exhibit and investigate a new candidate group that fails the unique units property but may satisfy the trivial unit property. No examples of groups with these properties are known to date.

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Heiko Dietrich, Melissa Lee, Andre Nies, Marc Vinyals. 2026-03-23. On the trivial units property and the unique product property. https://arxiv.org/abs/2603.22640

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