Parabolic subgroups and word problem in virtual Artin groups

We begin by establishing two fundamental results on standard parabolic subgroups of virtual Artin groups. We first show that a standard parabolic subgroup is naturally isomorphic to a virtual Artin group. Second, we prove that the intersection of two standard parabolic subgroups is a standard parabolic subgroup. Our main result is that, if all free of infinity standard parabolic subgroups of a given virtual Artin group VA[{\Gamma}] have a solvable word problem, then VA[{\Gamma}] itself has a solvable word problem. It follows that virtual Artin groups of FC type and, more generally, of affine-FC type, have a solvable word problem. We also prove that, if a virtual Artin group VA[{\Gamma}] has a solvable word problem, then the strong membership problem for any standard parabolic subgroup in VA[{\Gamma}] is solvable.