On the counterexamples to the unit conjecture for group rings

TL;DR

This paper discusses counterexamples to the Kaplansky unit conjecture, analyzing their determinants and explaining the existence of an infinite family of units in related group rings.

Contribution

It provides insights into the structure of units in group rings that counter the unit conjecture, including determinant analysis and infinite family characterization.

Findings

Counterexamples to the Kaplansky unit conjecture identified

Determinants of units in a specific matrix representation analyzed

Existence of a doubly infinite family of units explained

Abstract

We offer two comments on the beautiful papers of Giles Gardam and Alan Murray that yield counterexamples to the Kaplansky unit conjecture. First we discuss the determinants of these units in a certain $4\times 4$ matrix representation of the group ring. Then we explain why there is a doubly infinite family of units in the Murray paper.