Parabolic subgroups and word problem in virtual Artin groups

TL;DR

This paper studies the structure of virtual Artin groups, proving that their standard parabolic subgroups are themselves virtual Artin groups and that the word problem is solvable for certain types, with implications for subgroup membership problems.

Contribution

It establishes fundamental properties of standard parabolic subgroups in virtual Artin groups and links solvability of the word problem to subgroup structures.

Findings

Standard parabolic subgroups are isomorphic to virtual Artin groups.

Intersections of standard parabolic subgroups are standard parabolic.

Virtual Artin groups of FC and affine-FC types have solvable word problems.

Abstract

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.