On parabolic subgroups of Artin-Tits groups of spherical type

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.