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 length 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 combinatorial structures associated to $(m,n)$ which would yield counterexamples to both conjectures over the field $\mathbb{F}_2$ if they satisfy conditions $\mathsf{T}_1-\mathsf{T}_4$. Such combinatorial objects are called oriented product structures of type $(m,n)$. We also present a computer-search that can be utilized to search for counterexamples of a certain geometry by significantly pruning the search space. We prove that a class 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 $1\le m,n \le 13.$