[論文レビュー] Parabolic subgroups and word problem in virtual Artin groups

José Gálvez Mateos, Federica Gavazzi+1 more|arXiv (Cornell University)|Feb 27, 2026

Geometric and Algebraic Topology被引用数 0

ひとこと要約

The paper proves that standard parabolic subgroups of virtual Artin groups are themselves virtual Artin groups and establishes intersection and embedding properties, then shows solvability of the word problem for VA[Γ] from solvability on free-of-infinity subgroups, with applications to FC and affine-FC types.

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.

研究の動機と目的

• Establish that standard parabolic subgroups of VA[Γ] are naturally VA[Γ]s themselves.
• Show that intersections of standard parabolic subgroups are standard parabolic subgroups.
• Reduce solvability of the word problem for VA[Γ] to solvability on free-of-infinity standard parabolic subgroups.
• Apply results to obtain solvability of the word problem for VA[Γ] of FC type and affine-FC type.
• Investigate strong membership problems for standard parabolic subgroups under solvable word problem assumptions.

提案手法

• Prove VA[X][Γ] ≅ VA[ΓX] by constructing explicit isomorphisms.
• Show VAX[Γ] ∩ VAY[Γ] = VAX∩Y[Γ] for X,Y ⊆ S.
• Develop criteria linking solvability of the word problem on VA[ΓX] to that on VA[Γ], using strong membership problems.
• Use amalgamated product techniques to lift word problem solvability through decompositions (Theorem 2.18).
• Introduce and leverage structural decompositions of VA[Γ] as KVA[Γ] ⋊ W[Γ].
• Adapt known results on Coxeter and Artin subgroups to the virtual setting (Lemmas/Theorems 3.1–3.4).

実験結果

リサーチクエスチョン

• RQ1 Can standard parabolic subgroups of VA[Γ] be identified with VA[ΓX] via natural inclusions?
• RQ2 Is the intersection of two standard parabolic subgroups again a standard parabolic subgroup in VA[Γ]?
• RQ3 Does solvability of the word problem for all VA[ΓX] with X free of infinity imply solvability for VA[Γ]?
• RQ4 What is the impact of FC-type and affine-FC type on the word problem in VA[Γ]?
• RQ5 Is the strong membership problem for standard parabolic subgroups solvable under a solvable word problem for VA[Γ]?

主な発見

• VA[ΓX] is naturally isomorphic to the standard parabolic VAX[Γ].
• The intersection VAX[Γ] ∩ VAY[Γ] equals VAX∩Y[Γ].
• If VA[ΓX] has solvable word problem for all free-of-infinity X, then VA[Γ] has solvable word problem.
• Corollary: VA[Γ] has solvable word problem when Γ is of affine-FC type (and hence FC type).
• If VA[Γ] has a solvable word problem, then the strong membership problem for VAX[Γ] in VA[Γ] is solvable.
• Foundational results on strong membership for W[Γ] and A[Γ] in the respective ambient groups are established to support the main theorems.