论文标题
在自身形态组中的正常亚组上
On normal subgroups in automorphism groups
论文作者
论文摘要
我们描述了右角Artin组$ {\ rm aut}（a_γ）$的自动形态组中几乎可以解决的普通亚组的结构。特别是，我们证明了$ {\ rm aut}（a_γ）$的有限正常子组最多有两个，如果$γ$不是集团，则任何有限的普通亚组（$ {\ rm aut}（a_γ）$都是微不足道的。该属性对自动连续性和$ c^\ ast $ -Algebras具有影响：每个代数表达$φ\ colon l \ twheadHeadrightArrow {\ rm aut}（a_γ）（a_γ）（a_γ）$从本地紧凑的hausdorff group $ l $是连续的，并且仅是$a_γ$ n ies $ i是$ i是$ y是$ y是$ ysomor的， $ n \ geq 1 $。此外，如果$γ$不是一个联接，并且至少包含两个顶点，则在减少的组$ c^\ ast $ -Algebra of aut $（a_γ）$中，一组可逆元素的集合会密集。我们获得了$ {\ rm aut}（g_γ）$的类似结果，其中$g_γ$是环状组的图产品。此外，我们在定义图$γ$方面对Aut $（G_γ）$的中心进行了描述。
We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group ${\rm Aut}(A_Γ)$. In particular, we prove that a finite normal subgroup in ${\rm Aut}(A_Γ)$ has at most order two and if $Γ$ is not a clique, then any finite normal subgroup in ${\rm Aut}(A_Γ)$ is trivial. This property has implications to automatic continuity and to $C^\ast$-algebras: every algebraic epimorphism $φ\colon L\twoheadrightarrow{\rm Aut}(A_Γ)$ from a locally compact Hausdorff group $L$ is continuous if and only if $A_Γ$ is not isomorphic to $\mathbb{Z}^n$ for any $n\geq 1$. Further, if $Γ$ is not a join and contains at least two vertices, then the set of invertible elements is dense in the reduced group $C^\ast$-algebra of Aut$(A_Γ)$. We obtain similar results for ${\rm Aut}(G_Γ)$ where $G_Γ$ is a graph product of cyclic groups. Moreover, we give a description of the center of Aut$(G_Γ)$ in terms of the defining graph $Γ$.
