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Parabolic subgroups and word problem in virtual Artin groups

José Gálvez Mateos, Federica Gavazzi, Luis Paris

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[Γ] have a solvable word problem, then VA[Γ] 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[Γ] has a solvable word problem, then the strong membership problem for any standard parabolic subgroup in VA[Γ] is solvable.

Parabolic subgroups and word problem in virtual Artin groups

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[Γ] have a solvable word problem, then VA[Γ] 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[Γ] has a solvable word problem, then the strong membership problem for any standard parabolic subgroup in VA[Γ] is solvable.
Paper Structure (8 sections, 29 theorems, 23 equations, 1 figure)

This paper contains 8 sections, 29 theorems, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Standard parabolic subgroups of Coxeter groups
  4. Standard parabolic subgroups of Artin groups
  5. Virtual Artin groups
  6. Amalgamated products
  7. Standard parabolic subgroups of virtual Artin groups
  8. Word problem

Key Result

Theorem 1.1

Let $X \subseteq S$. The natural homomorphism ${\rm VA} [\Gamma_X] \to {\rm VA}_X [\Gamma]$ that maps $\sigma_x$ to $\sigma_x$ and $\tau_x$ to $\tau_x$ for all $x \in X$ is an isomorphism.

Figures (1)

  • Figure 3. 1: Coxeter graph $A_2$

Theorems & Definitions (57)

  • Definition
  • Definition
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • proof
  • Theorem 1.6
  • Theorem 2.1: Tits Tit69
  • ...and 47 more