Ends of groups: a nonstandard perspective

TL;DR

This paper introduces a nonstandard analysis approach to the concept of ends in proper geodesic metric spaces and applies it to Cayley graphs of finitely generated groups, providing new proofs and insights.

Contribution

It develops a nonstandard framework for understanding ends of groups and extends this approach to relatively Cayley graphs, offering novel proofs of classical results.

Findings

Nonstandard treatment of ends of metric spaces

New proofs of fundamental results on ends of groups

Extension to ends of relatively Cayley graphs

Abstract

We give a nonstandard treatment of the notion of ends of proper geodesic metric spaces. We then apply this nonstandard treatment to Cayley graphs of finitely generated groups and give nonstandard proofs of many of the fundamental results concerning ends of groups. We end with an analogous nonstandard treatment of the ends of relatively Cayley graphs, that is Cayley graphs of cosets of finitely generated groups.