Right-angled Coxeter quandles and polyhedral products

Abstract

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