Right-angled Coxeter quandles and polyhedral products

TL;DR

This paper explores the structure of a group associated with Coxeter groups, showing that for right-angled cases, its classifying space can be described using polyhedral products, with applications to understanding its topology.

Contribution

It establishes a new connection between Coxeter quandles, their associated groups, and polyhedral products for right-angled Coxeter groups.

Findings

The group $ ext{Ad}(X_W)$ is a pullback involving $W$.

The classifying space of $ ext{Ad}(X_W)$ is a polyhedral product for right-angled Coxeter groups.

Applications demonstrate the utility of this topological description.

Abstract

To a Coxeter group $W$ one associates a quandle $X_W$ from which one constructs a group $\mathrm{Ad}(X_W)$. This group turns out to be an intermediate object between $W$ and the associated Artin group. By using a result of Akita, we prove that $\mathrm{Ad}(X_W)$ is given by a pullback involving $W$, and by using this pullback, we show that the classifying space of $\mathrm{Ad}(X_W)$ is given by a space called a polyhedral product whenever $W$ is right-angled. Two applications of this description of the classifying space are given.