Normalisers of parabolic subgroups of Artin--Tits groups and Tits cone intersections

Let $\mathrm{\Gamma }$ be a Coxeter diagram and let $J\subseteq \mathrm{\Gamma }$. Motivated by 3-fold flops, Iyama and Wemyss study the hyperplane arrangement in the Tits cone intersection of $J$, which is a $J$-relative generalisation of the classical Coxeter arrangement. For $\mathrm{\Gamma }$ of finite-type, we show that its complexified hyperplane complement is a $K(\pi ,1)$ space for the normaliser (quotient) of the standard parabolic subgroup of the Artin–Tits group attached to $J$. For general $\mathrm{\Gamma }$ we show that Brink–Howlett’s groupoid, which describes normalisers of parabolic subgroups of Coxeter groups, has its universal cover described by the wall-and-chamber structure of the Tits cone intersection. We use this to show that wall crossing sequences satisfy an “atomic Matsumoto relation”, generalising a theorem of Ko and answering questions raised by Iyama and Wemyss.