Fuglede's conjecture holds on $\mathbb{Z}_p^2 \times \mathbb{Z}_q$

TL;DR

This paper proves Fuglede's conjecture for the group _{p}^2 imes _{q} by employing a novel approach rooted in discrete geometry, extending previous results on elementary abelian groups.

Contribution

It introduces a new method based on discrete geometry to establish Fuglede's conjecture on _{p}^2 imes _{q}, filling a gap in the understanding of the conjecture on product groups.

Findings

Fuglede's conjecture holds on _{p}^2 imes _{q}.

Developed a geometric method applicable to product groups.

Extended the class of groups where the conjecture is verified.

Abstract

The study of Fuglede's conjecture on the direct product of elementary abelian groups was initiated by Iosevich et al. For the product of two elementary abelian groups the conjecture holds. For $\mathbb{Z}_p^3$ the problem is still open if $p\ge 11$. In connection we prove that Fuglede's conjecture holds on $\mathbb{Z}_{p}^2\times \mathbb{Z}_q$ by developing a method based on ideas from discrete geometry.