论文标题
算术近似晶格的较高有限性属性：数字字段的等级定理
Higher finiteness properties of arithmetic approximate lattices: The Rank Theorem for number fields
论文作者
论文摘要
我们通过粗糙的几何形状引入了可数近似基团的几何和同源性特性，然后研究这些有限性的属性属性。对于没有无限位置的s-砷近似组，我们表明有限长度是有限的，并且明确计算了此有限长度。在简单的情况下，它比本地等级的总和少。这将Bux，Köhl和第二作者的等级定理从正常特征到特征零。我们的证明是基于其证明的几何版本，但除了还原理论的某些投入外，它是自由的。这表明特征为零的算术组和有关有限性能的积极特征的明显差异完全是由于无限位置的存在。
We introduce geometric and homological finiteness properties for countable approximate groups via coarse geometry and then study these finiteness properties for S-arithmetic reductive approximate groups. For S-arithmetic approximate groups without infinite places we show that the finiteness length is finite and compute this finiteness length explicitly. In the simple case it is one less than the sum of the local ranks. This extends the Rank Theorem of Bux, Köhl and the second author from positive characteristic to characteristic zero. Our proof is based on a geometric version of their proof, but except for some input from reduction theory it is characteristic free. This indicates that the apparent differences between arithmetic groups in characteristic zero and positive characteristic concerning finiteness properties are entirely due to the presence of infinite places.
