Beyond graph products and cactus groups: quandle products of groups

TL;DR

This paper introduces quandle products of groups, a new family encompassing several known group constructions, and explores their geometric properties using quasi-median Cayley graphs.

Contribution

It defines quandle products of groups, demonstrates their geometric structure, and connects them to existing group constructions like graph products and cactus groups.

Findings

Quandle products admit quasi-median Cayley graphs.

The geometric approach yields insights into the structure of quandle products.

The framework unifies various group constructions under a common geometric perspective.

Abstract

In this paper, we introduce and initiate the study of quandle products of groups, a family of groups that includes graph products of groups, cactus groups, wreath products, and the recently introduced trickle groups. Our approach is geometric: we show that quandle products admit quasi-median Cayley graphs; and, then, we exploit this geometry to deduce various valuable information about quandle products.