论文标题
类型$ f_4 $和$ h_4 $类型类型的Artin Groups与$ d_4 $的类型不符
Artin groups of types $F_4$ and $H_4$ are not commensurable with that of type $D_4$
论文作者
论文摘要
在最近的一篇文章中，Cumplido和Paris研究了球形类型的Artin群体之间的可怜性问题。他们的分析留下了六个案例，对于以下一对Artin组：$（f_4，d_4）$，$（H_4，D_4）$，$（f_4，h_4）$，$（e_6，d_6，d_6）$，$（E_7，d_7，d_7）$和$（e_8，d_8，d_8，d_8）$。在本说明中，我们解决了这些情况的前两种，即，我们表明类型$ f_4 $和$ h_4 $类型的Artin组与$ d_4 $的类型不符。作为关键步骤，我们意识到Artin类型$ d_4 $的抽象要素是带有三个穿刺的圆环的扩展映射类组。我们还找到了Artin类型$ d_4 $的Artin组的自动形态组，并在所有不可约束的球形类型模型中的扭力元素，其订单和共轭类别的描述。
In a recent article, Cumplido and Paris studied the question of commensurability between Artin groups of spherical type. Their analysis left six cases undecided, for the following pairs of Artin groups: $(F_4,D_4)$, $(H_4,D_4)$, $(F_4,H_4)$, $(E_6,D_6)$, $(E_7,D_7)$, and $(E_8,D_8)$. In this note we resolve the first two of these cases, namely, we show that the Artin groups of types $F_4$ and $H_4$ are not commensurable with that of type $D_4$. As a key step, we realize the abstract commensurator of the Artin group of type $D_4$ as the extended mapping class group of the torus with three punctures. We also find the automorphism group of the Artin group of type $D_4$ and obtain a description of torsion elements, their orders and conjugacy classes in all irreducible Artin groups of spherical type modulo their centers.
