On quasi-isometry and choice

TL;DR

This paper demonstrates that the symmetry property of quasi-isometry relations in hyperbolic spaces implies the axiom of choice, linking geometric concepts to foundational set theory principles.

Contribution

It establishes a novel connection between the symmetry of quasi-isometry and the axiom of choice, even in restricted geometric settings.

Findings

Symmetry of quasi-isometry implies the axiom of choice.

The result is sharp; more restrictive settings do not imply choice.

The Bottleneck Theorem also implies the axiom of choice.

Abstract

In this note we prove that the symmetry of the quasi-isometry relation implies the axiom of choice, even when the relation is restricted to geodesic hyperbolic spaces. We show that this result is sharp by demonstrating that symmetry of quasi-isometry in an even more restrictive setting does not imply the axiom of choice. The "Bottleneck Theorem" of Jason Fox Manning also implies choice.