Coarse Freundenthal compactification and ends of groups

Abstract

A coarse compactification of a proper metric space $X$ is any compactification of $X$ that is dominated by its Higson compactification. In this paper we describe the maximal coarse compactification of $X$ whose corona is of dimension $0$. In case of geodesic spaces $X$, it coincides with the Freundenthal compactification of $X$. As an application we provide an alternative way of extending the concept of the number of ends from finitely generated groups to arbitrary countable groups. We present a geometric proof of a generalization of Stallings' theorem by showing that any countable group of two ends contains an infinite cyclic subgroup of finite index. Finally, we define ends of arbitrary coarse spaces.