On fusion control in FC type Artin-Tits groups

Abstract

We consider Artin-Tits groups of FC type and prove the two following results. If any two distinct elements of a standard parabolic subgroup are conjugated by a standard generator of the whole group, then this generator has to be in the subgroup. We also prove that classical transversals of standard parabolic subgroups of Artin-Tits groups of FC type are compatible with intersection with standard parabolic subgroups: If AX,AY are two standard parabolic subgroups of an Artin-Tits groups A of FC type and T is the classical transversal of AX in A, then T∩AY is a transversal of AX∩Y in AY. So the two properties hold for Artin-Tits groups of spherical type and Right-angled Artin-Tits groups.