On the trivial units property and the unique product property

TL;DR

This paper presents computational experiments on the trivial units and unique product properties in group rings of torsion-free groups, exploring their relation to Kaplansky's conjectures and identifying new candidate groups.

Contribution

It provides the first classification of certain symmetric units in the Hantzsche-Wendt group and introduces a new candidate group with unique properties.

Findings

Classified symmetric non-trivial units in the Hantzsche-Wendt group

Identified a candidate group failing the unique units property

Proposed a group potentially satisfying the trivial unit property

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.