Ends of spaces via linear algebra

TL;DR

This paper develops a linear algebra-based framework to analyze the ends of spaces as points at infinity, unifying various existing notions of ends through eigensets of linear operators on Boolean algebras.

Contribution

It introduces a novel approach using linear algebra to trim Boolean algebras of sets, unifying and generalizing existing concepts of ends of spaces.

Findings

Ends can be characterized as eigensets of linear operators.

The framework unifies multiple notions of ends in a single algebraic setting.

The approach provides a new perspective on the behavior of spaces at infinity.

Abstract

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.