On zero-divisors and units in group rings of torsion-free CAT$(0)$ groups

Abstract

This paper addresses two of Kaplansky's conjectures concerning group rings $K[G]$, where $K$ is a field and $G$ is a torsion-free group: the zero-divisor conjecture, which asserts that $K[G]$ has no non-trivial zero-divisors, and the unit conjecture, which asserts that $K[G]$ has no non-trivial units. While the zero-divisor conjecture still remains open, the unit conjecture was disproven by Gardam in 2021. The search for more counterexamples remains an open problem. Let $m$ and $n$ be the cardinality of support of two non-trivial elements $\alpha, \beta \in \mathbb{F}_2[G]$, respectively. We address these conjectures by introducing a process called \text{left alignment} and recursively constructing the taikos of size $(m,n)$ which would yield counterexamples to both conjectures over the field $\mathbb{F}_2$ if they satisfy conditions $\mathsf{T}_1-\mathsf{T}_4$ given in \cite{Mineyev2024}. We also present a computer-search method that can be utilized to search for counterexamples of a certain geometry by significantly pruning the search space. We prove that a class of CAT(0) groups with certain geometry cannot be counterexamples to these conjectures. Moreover, we prove that for $ 1\le m \le 5$ and $n$ any positive integer, there are no counterexamples to the conjectures such that the associated oriented product structures are of type $(m,n)$. With the aid of computer, we prove that, in fact, there are no such counterexamples of the length combination $(m,n)$ where $1\le m \le 13$ and $1\le n \le 13$.