On Gardam's and Murray's units in group rings

TL;DR

This paper demonstrates that units in torsion-free group rings identified by Gardam are twisted unitary elements, providing new examples and insights into their symmetry properties, thereby clarifying aspects of Gardam's construction.

Contribution

It establishes that Gardam's units are twisted unitary elements, justifies certain construction choices, and expands the known examples of units in group rings.

Findings

Units are twisted unitary elements

All known units exhibit non-trivial symmetry

Provides additional examples of units in group rings

Abstract

We show that the units found in torsion-free group rings by Gardam are twisted unitary elements. This justifies some choices in Gardam's construction that might have appeared arbitrary, and yields more examples of units. We note that all units found up to date exhibit non-trivial symmetry.