On Common Divisors of Fox Derivatives with towards to Zero Divisors of Group Rings

TL;DR

This paper constructs specific group presentations using the Composition--Diamond Lemma to show that certain Fox derivatives share common divisors, implying the existence of zero divisors in the group ring if the second homotopy group is nontrivial.

Contribution

It introduces a method to construct groups with Fox derivatives sharing common divisors, linking topological properties to algebraic zero divisors in group rings.

Findings

Constructed group presentations with common divisors in Fox derivatives.

Established a connection between second homotopy group and zero divisors in group rings.

Provided a new approach to study zero divisors via group presentations.

Abstract

Using Composition--Diamond Lemma we construct presentations of groups $G = \langle x_1,\ldots,x_n \, | \, r_1,\ldots, r_m \rangle$ with the following property; for a fixed $1 \le i \le n$, and for all $1 \le j \le m$, Fox derivatives $\partial r_j / \partial x_i$ have common divisor. It follows that if $\pi_2(K) \ne 0$, where $K$ is the standard $2$-complex associated with $G$ then the group ring $\mathbb{Z}[G]$ has nontrivial zero divisors.