Given an Artin group A_Γ, a common strategy in the study of A_Γ is the reduction to parabolic subgroups whose defining graphs have small diameter, i.e., showing that A_Γ has a specific property if and only if all “small” parabolic subgroups of A_Γ have this property. Since “small” parabolic subgroups are the building blocks of A_Γ one needs to study their behavior, in particular their intersections. The conjecture we address here says that the class of parabolic subgroups of A_Γ is closed under intersection. Under the assumption that intersections of parabolic subgroups in complete Artin groups are parabolic, we show that the intersection of a complete parabolic subgroup with an arbitrary parabolic subgroup is parabolic. Further, we connect the intersection behavior of complete parabolic subgroups of A_Γ to fixed point properties and to automatic continuity of A_Γ using Bass–Serre theory and a generalization of the Deligne complex.

给定一个 Artin 群A _Г，研究A _Г的常用策略是还原为其定义图具有小直径的抛物线子群，即，表明A _Г具有特定属性当且仅当所有“小”抛物线子群A _Г有这个性质。由于“小”抛物线子群是A _Γ的构建块，因此需要研究它们的行为，特别是它们的交集。我们在这里提出的猜想表明， A _Γ的抛物线子群类在交集下是闭的。假设完全Artin群中的抛物线子群的交是抛物线的，我们证明完全抛物线子群与任意抛物线子群的交是抛物线的。此外，我们使用 Bass-Serre 理论和 Deligne 复形的推广将A _Γ的完全抛物线子群的交行为与不动点属性和A _{Γ的自动连续性连接起来。}