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Quasi-isometric rigidity of the integers: an elementary primer

Tarik Aougab, Hikaru Jitsukawa, Kim Ruane

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.

Quasi-isometric rigidity of the integers: an elementary primer

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.
Paper Structure (12 sections, 11 theorems, 41 equations, 15 figures)

This paper contains 12 sections, 11 theorems, 41 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Quasi-Isometry
  3. The set-up
  4. Step 1: Passing from $\Gamma$ to $\mathbb{R}$
  5. Step 2: An Action on the Ends
  6. Step 3: Finding a Copy of $\mathbb{Z}$
  7. Appendix A: Alternative Proof Methods
  8. Appendix B: Extending to Groups of Linear Growth
  9. Appendix C: Flame's Math Art

Key Result

Theorem A

Let $G$ be a finitely generated group that is quasi-isometric to $\mathbb{R}$; then $G$ is virtually $\mathbb{Z}$.

Figures (15)

  • Figure 1: Two Cayley Graphs for $\mathbb{Z}$
  • Figure 2: Quasi-Isometry
  • Figure 3: The map $*$ applied to an element of $\mathbb{R}$
  • Figure 4: The inductive step of Lemma \ref{['lem:ActionOnFarIntervals']}
  • Figure 5: The contradiction in Lemma \ref{['lem:NoFlip']}. The points $x$ and $x+n'$, which have large distance in the domain, are mapped close to each other in the codomain, contradicting the fact that $g*$ is a quasi-isometry.
  • ...and 10 more figures

Theorems & Definitions (28)

  • Theorem A
  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Definition 4
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 18 more