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

TL;DR

This paper investigates Kaplansky's zero-divisor and unit conjectures in group rings of torsion-free CAT(0) groups, introducing new methods and computational searches to identify potential counterexamples, and proving their absence in certain cases.

Contribution

It introduces a recursive process called left alignment and a computer-search method to find counterexamples, proving their non-existence for specific parameters and group geometries.

Findings

No counterexamples for support sizes up to (13,13).

Certain CAT(0) groups cannot produce counterexamples.

The left alignment process aids in systematic search for counterexamples.

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 $1\le n \le 13$.