On parabolic subgroups of Artin-Tits groups of spherical type

TL;DR

This paper proves that parabolic subgroups in Artin-Tits groups of spherical type form a lattice under inclusion, extending known results from braid groups and introducing a new complex analogous to the curve complex.

Contribution

It establishes the lattice structure of parabolic subgroups and introduces a simplicial complex of irreducible parabolic subgroups as a new geometric tool.

Findings

Intersection of two parabolic subgroups is parabolic

The set of parabolic subgroups forms a lattice

A new simplicial complex of irreducible parabolic subgroups is proposed

Abstract

We show that, in an Artin-Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin-Tits groups of spherical type a result that was previously known for braid groups. To obtain the above results, we show that every element in an Artin-Tits group of spherical type admits a unique minimal parabolic subgroup containing it. Also, the subgroup associated to an element coincides with the subgroup associated to any of its powers or roots. As a consequence, if an element belongs to a parabolic subgroup, all its roots belong to the same parabolic subgroup. We define the simplicial complex of irreducible parabolic subgroups, and we propose it as the analogue, in Artin-Tits groups of spherical type, of the celebrated complex of curves which is an important tool in braid groups, and more generally in mapping class groups. We conjecture that the complex of irreducible parabolic subgroups is $\delta$-hyperbolic.