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

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 $$0$$.

MSC:

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

Keywords:

group ring; unit conjecture
Full Text:

References:

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