A counterexample to the unit conjecture for group rings. (English) Zbl 1494.16026

Gardam, Giles

Ann. Math. (2) 194, No. 3, 967-979 (2021).

A long standing conjecture known as Kaplansky’s unit conjecture stated that in the group algebra \(KG\) of a torsion-free group \(G\) over a field \(K\) each unit is trivial, i.e., of the shape \(kg\) for some \(k \in K \setminus \{ 0 \}\) and \(g \in G\). A counterexample to this conjecture is given in this paper.
Kaplansky’s conjecture is known to hold for groups having unique products, where \(G\) has unique products, if for any non-empty finite subsets \(A,B \subset G\) the set \(A \cdot B = \{ab \ | \ a \in A, b \in B \}\) contains an element which is uniquely expressible as \(ab\) for \(a \in A\) and \(b \in B\). Several examples of group without the unique product property are known, the most elementary one having been discovered by Promislow and known as the Promislow group or Hantzsche-Wendt group or the Fibonacci group \(F(2,6)\). It is defined as \[ P = \langle a,b \ | \ b^{-1}a^2b = a^{-2}, a^{-1}b^2a = b^{-2} \rangle. \] For \(K\) the field with \(2\) elements an explicit element in \(KP\) is presented which is shown to be a non-trivial unit by elementary calculations. The existence of the unit raises the question on how the whole unit group of \(KP\) might look like and the author shows that this group is torsion-free, linear, not finitely generated and contains free non-abelian subgroups. Two questions are raised at the end: is having unique products for a group equivalent to being diffuse? Is having unique products for a torsion-free group equivalent to satisfying Kaplansky’s conjecture?
The group \(P\) is known to satisfy the other two famous conjectures of Kaplansky on group rings – the zero divisor conjecture and the idempotent conjecture – and these remain open. After this paper had appeared as a preprint A. Murray produced variations of the unit in \(KP\) also for any other field \(K\) of positive characteristic [A. Murray, “More counterexamples to the unit conjecture for group rings”, Preprint, arXiv:2106.02147]. Kaplansky’s unit conjecture remains open in characteristic \(0\).

Reviewer: Leo Margolis (Brussel)

MSC:

16S34	Group rings
16U60	Units, groups of units (associative rings and algebras)

Keywords:

group ring; unit conjecture

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