论文标题
在Bieri-Neumann-Strebel-Renz $σ$ -Invariants上
On the Bieri-Neumann-Strebel-Renz $Σ$-invariants of the Bestvina-Brady groups
论文作者
论文摘要
我们研究Bieri-Neumann-Strebel-Renz不变式，我们证明了以下标准：对于$ h $ and $ h $和$ k $的类型$ fp_n $，使得$ [h，h] \ subseteq k \ subseteq h $ subseteq h $ use $χ： σ^n（k，k，\ mathbb {z}）$，仅当每个字符$μ：h \ to \ to $ [μ] \ inσ^n（h，\ mathbb {z}）$ for每个字符$μ：h \ to \ mathbb {r} $扩展了$χ$。当$ k $和$ h $是类型$ f_n $的组时，同质不变的$σ^n（ - ）$也相同。我们使用这些标准来完成对Bieri-Stallings组的$σ$ invariants $ g_m $的描述，更一般地描述了Bestvina-Brady群体的$σ$ -Invariants。我们还表明，当我们仅假设$ k $是$ h $的亚正常亚组时，只有此类标准的“仅”方向存在，其中两组都是$ fp_n $的类型。我们将最后一个结果应用于花圈产品。
We study the Bieri-Neumann-Strebel-Renz invariants and we prove the following criterion: for groups $H$ and $K$ of type $FP_n$ such that $[H,H] \subseteq K \subseteq H$ and a character $χ: K \to \mathbb{R}$ with $χ([H,H]) = 0$ we have $[χ] \in Σ^n(K, \mathbb{Z})$ if and only if $[μ] \in Σ^n(H, \mathbb{Z})$ for every character $μ: H \to \mathbb{R}$ that extends $χ$. The same holds for the homotopical invariants $Σ^n(-)$ when $K$ and $H$ are groups of type $F_n$. We use these criteria to complete the description of the $Σ$-invariants of the Bieri-Stallings groups $G_m$ and more generally to describe the $Σ$-invariants of the Bestvina-Brady groups. We also show that the "only if" direction of such criterion holds if we assume only that $K$ is a subnormal subgroup of $H$, where both groups are of type $FP_n$. We apply this last result to wreath products.
