基于线性代数的空间端
Ends of spaces via linear algebra
摘要：
我们发展了一个理论，可以被认为是粗糙理论的前传。我们把空间的两端看作是无穷远处的额外点。为了讨论空间在无穷远处的行为，我们需要一个逼近无穷远处的概念（一个测度）。最简单的方法是列出$X$的有界子集（即远离无穷大），并且该列表应满足某些基本性质。这样的列表$S_X$我们称之为集合$X$上的\textbf{scale}（见第3节）。为了使用斯通对偶定理的思想，我们考虑包含$S_X$的$X$的幂集$2^X$的子布尔代数$BA_X$，并且自然地导致\textbf{scaled Boolean algebra} $(X,S_X,BA_X)$的端点的概念，该端点可以附加到$X$并且形成新的缩放布尔代数$(\bar X,S_X,\overline{BA_X})$，即\textbf{compact at infinity}。给定一个缩放空间$(X,S_X)$，最自然的缩放布尔代数是$(X,S_X,2^X)$，它可能离$X$的几何太远了。因此，我们需要弄清楚如何将$2^X$修剪成更小的子布尔代数$BA_X$。更一般地说，如何修剪一个子布尔代数$BA_X$到一个更小的。这是利用线性代数的思想完成的。也就是说，我们考虑$BA_X$上自然产生的$S_X$ -线性算子族$\mathcal{F}$，较小的子布尔代数$BA_{\mathcal{F}}$由$\mathcal{F}$的特征集组成，$\mathcal{F}$是线性代数的特征向量的模拟.我们表明，文献中定义的所有结束到目前为止（弗罗因登塔尔结束，结束的生成群，斯派克结束，Cornulier结束，结束粗糙的空间）是这样一个过程的特殊情况。
英文摘要：
We develop a theory that may be considered as a prequel to the coarse theory. We are viewing ends of spaces as extra points at infinity. In order to discuss behaviour of spaces at infinity one needs a concept (a measure) of approaching infinity. The simplest way to do so is to list subsets of $X$ that are bounded (i.e. far from infinity) and that list should satisfy certain basic properties. Such a list $S_X$ we call a \textbf{scale} on a set $X$ (see Section 3). In order to use ideas from the Stone Duality Theorem we consider sub-Boolean algebras $BA_X$ of the power set $2^X$ of $X$ that contain $S_X$ and that leads naturally to the concept of ends of a \textbf{scaled Boolean algebra} $(X,S_X,BA_X)$ which can be attached to $X$ and form a new scaled Boolean algebra $(\bar X,S_X,\overline{BA_X})$ that is \textbf{compact at infinity}. Given a scaled space $(X,S_X)$ the most natural scaled Boolean algebra is $(X,S_X,2^X)$ which can be too far removed from the geometry of $X$. Therefore we need to figure out how to trim $2^X$ to a smaller sub-Boolean algebra $BA_X$. More generally, how to trim a sub-Boolean algebra $BA_X$ to a smaller one. That is done using ideas from linear algebra. Namely, we consider a family $\mathcal{F}$ of naturally arising $S_X$-linear operators on $BA_X$ and the smaller sub-Boolean algebra $BA_{\mathcal{F}}$ consists of eigensets of $\mathcal{F}$, an analog of eigenvectors from linear algebra. We show that all ends defined in literature so far (Freundenthal ends, ends of finitely generated groups, Specker ends, Cornulier ends, ends of coarse spaces) are special cases of such a process.
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