Ends of large scale groups

TL;DR

This paper unifies the theory of ends for finitely generated and topological groups by defining ends for coarse spaces, and extends key results like the Svarc-Milnor Lemma to large scale groups.

Contribution

It introduces a generalized notion of ends for coarse spaces and large scale groups, bridging existing theories and proving new properties like hyperbolic groups having finite asymptotic dimension.

Findings

Groups with finitely many ends have at most two ends.

Large scale hyperbolic groups with bounded geometry have finite asymptotic dimension.

A version of the Svarc-Milnor Lemma is established for 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.