[论文解读] 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 0

一句话总结

论文证明虚拟Artin群的标准类 Parabolic 子群本身也是虚拟Artin群，并确立交叉和嵌入性质；随后从可解无穷大自由子群的可解性推出 VA[Γ] 的词问题可解性，并应用于 FC 类型和仿 FC 类型。

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.

研究动机与目标

• 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.

提出的方法

• 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).

实验结果

研究问题

• 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]?

主要发现

• 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.