Normalisers of parabolic subgroups of Artin--Tits groups and Tits cone intersections
Owen Garnier, Edmund Heng, Anthony Licata, Oded Yacobi
TL;DR
The paper develops a relative framework tying hyperplane arrangements from Tits cone intersections to normalisers of parabolic subgroups in Artin–Tits groups. It proves that, for finite-type Coxeter data, the complexified J-cone arrangement yields a $K(\\pi,1)$ space for the normaliser quotient $N(A,J)$, and in general constructs a geometric realization of Brink–Howlett’s groupoid via wall–and–chamber decompositions, leading to an atomic Matsumoto-type relation. By comparing the Deligne groupoid with the Brink–Howlett and reduced ribbon groupoids, the authors produce an explicit equivalence $\\mathscr{D}/N(W,J) \\cong \\mathscr{R}$ and derive algebraic descriptions of the fundamental groups of the related complexified hyperplane complements. Consequently, when $W$ is finite, $N(A,J)$ and $N(P,J)$ admit $K(\\pi,1)$ realizations as fundamental groups of quotients of hyperplane complements, with tangible consequences for cohomology presentations via Orlik–Solomon data. These results extend Matsumoto-type relations beyond finite-type settings and answer questions of Iyama–Wemyss about atomic wall-crossing phenomena in the relative parabolic context.
Abstract
Let $Γ$ be a Coxeter diagram and let $J \subseteq Γ$. 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 $Γ$ of finite-type, we show that its complexified hyperplane complement is a $K(π,1)$ space for the normaliser (quotient) of the standard parabolic subgroup of the Artin--Tits group attached to $J$. For general $Γ$ 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.