论文标题
$σ$ -Invariants $ s $ -arithmetic子组的Borel组
The $Σ$-invariants of $S$-arithmetic subgroups of Borel groups
论文作者
论文摘要
给定一个经典类型的Chevalley组$ \ MATHCAL {G} $和Borel子组$ \ Mathcal {B} \ subseteq \ Mathcal {G} $，我们计算$ s $ s $ s $ s $ arithmetic ofter-arithmetic ofter-arithmetic ofter-arithmetic ofter-arithmetic群$ \ mathcal的$ s $ \ bb bb b} $ n） $ n $是足够大的产品的产物。为此，我们让$ \ Mathcal {b}（\ Mathbb {z} [1/n]）$在欧几里得建筑物$ x $上行动，该$ x $由bruhat-tits buildings $ x_p $相关的产品与$ \ nathcal {g} $相关的产品，其中$ p $ cu $ cum and prime n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $。在证明过程中，我们在$ \ mbox {cat（cat（0）} $ - 空格上引入了必要的和充分的条件，以保持连续。我们将这些条件应用于Infinity $τ\ subset \ partial _ {\ infty} x $其所谓的抛物线构建$ x^τ$的每个单纯子，我们从几何学的角度进行研究。此外，我们介绍了组合摩尔斯理论中的新技术，这使我们能够利用基本$ n $连接性的概念，而不是实际的$ n $连接性。我们的大多数建筑理论结果在球形和欧几里得建筑的一般框架中得到了证明。例如，我们证明，如果$δ$足够厚，并且$ \ mbox {aut}（δ）$在$Δ$上进行过渡性，则$δ$的$δ$相对的每个房间对面的每个腔室都包含公寓。
Given a Chevalley group $\mathcal{G}$ of classical type and a Borel subgroup $\mathcal{B} \subseteq \mathcal{G}$, we compute the $Σ$-invariants of the $S$-arithmetic groups $\mathcal{B}(\mathbb{Z}[1/N])$, where $N$ is a product of large enough primes. To this end, we let $\mathcal{B}(\mathbb{Z}[1/N])$ act on a Euclidean building $X$ that is given by the product of Bruhat--Tits buildings $X_p$ associated to $\mathcal{G}$, where $p$ runs over the primes dividing $N$. In the course of the proof we introduce necessary and sufficient conditions for convex functions on $\mbox{CAT(0)}$-spaces to be continuous. We apply these conditions to associate to each simplex at infinity $τ\subset \partial_{\infty} X$ its so-called parabolic building $X^τ$, which we study from a geometric point of view. Moreover, we introduce new techniques in combinatorial Morse theory, which enable us to take advantage of the concept of essential $n$-connectivity rather than actual $n$-connectivity. Most of our building theoretic results are proven in the general framework of spherical and Euclidean buildings. For example, we prove that the complex opposite each chamber in a spherical building $Δ$ contains an apartment, provided $Δ$ is thick enough and $\mbox{Aut}(Δ)$ acts chamber transitively on $Δ$.
