Quasi-isometric rigidity of the integers: an elementary primer

TL;DR

This paper provides an elementary, accessible proof of the quasi-isometric rigidity of the integers, showing that any finitely generated group quasi-isometric to the integers is virtually the integers, aimed at first-year students.

Contribution

It offers a simple, elementary proof of a classical theorem in geometric group theory, making the result more accessible for beginners.

Findings

Any finitely generated group quasi-isometric to the integers is virtually the integers.

The proof is designed to be accessible for first-year students.

The paper emphasizes elementary methods in geometric group theory.

Abstract

Chatawate (Flame) Ruethaimetapat was a passionate, enthusiastic, and wonderful person who passed away in August of 2024. At the time of their passing they were working towards their PhD, specializing in geometric group theory. Flame was just as excited about learning new mathematics as they were about sharing it with everyone else, so it's no surprise that they spent a lot of time thinking about how to write down expository proofs of classical theorems that would be accessible for first year students. In particular, they sought a simple, elementary proof of the fact that any finitely generated group quasi-isometric to the integers is virtually the integers. In the spirit of this endeavor and in loving memory of Flame, we present such a proof here.