Ends of large scale groups

Abstract

The aim of this paper is to unify the theory of ends of finitely generated groups with that of ends of locally compact, metrizable and connected topological groups. In both theories one proves that, if the number of ends is finite, then it must be at most $2$. In both theories groups of two ends are characterized as having an infinite cyclic subgroup of either finite index or such that its coset space is compact. Our generalization amounts to defining the space of ends of any coarse space and then applying it to large scale groups, a class of groups generalizing both finitely generated groups and locally compact, metrizable and connected topological groups. Additionally, we prove a version of Svarc-Milnor Lemma for large scale groups and we prove that coarsely hyperbolic large scale groups have finite asymptotic dimension provided they have bounded geometry.