论文标题
Artin组的渐近维度和Coxeter组的新上限
Asymptotic dimension of Artin groups and a new upper bound for Coxeter groups
论文作者
论文摘要
如果$a_γ$（$w_γ$）是Artin（Coxeter）组，其定义图$γ$我们用$ sim（γ）$表示$γ$的最大集团的顶点数量。我们表明，如果$ sim（γ）= 2 $，则$asdima_γ\ leq sim（γ）$。我们猜想不平等对每个Artin群体都有。我们证明，如果对于所有猜想所拥有的无限ARTIN（Coxeter）组，则它将为所有Artin（Coxeter）组。作为推论，我们表明所有Coxeter组的$ASDIMW_γ\ leq SIM（γ）$，这是Coxeter组的渐近维度最著名的上限。作为进一步的推论，我们表明，任何具有$ SIM（γ）= 3 $的大型ARTIN组的渐近维度正好是两个。
If $A_Γ$ ($W_Γ$) is the Artin (Coxeter) group with defining graph $Γ$ we denote by $Sim(Γ)$ the number of vertices of the largest clique in $Γ$. We show that $asdimA_Γ\leq Sim(Γ)$, if $Sim(Γ)=2$. We conjecture that the inequality holds for every Artin group. We prove that if for all free of infinity Artin (Coxeter) groups the conjecture holds, then it holds for all Artin (Coxeter) groups. As a corollary, we show that $asdimW_Γ\leq Sim(Γ)$ for all Coxeter groups, which is the best known upper bound for the asymptotic dimension of Coxeter Groups. As a further corollary, we show that the asymptotic dimension of any Artin group of large type with $Sim(Γ)=3$ is exactly two.
