Title: On parabolic subgroups of Artin-Tits groups of spherical type,
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.