Are free groups of different ranks bi-invariantly quasi-isometric?

TL;DR

This paper proves that free groups of different finite ranks are not bi-invariantly quasi-isometric, establishing that only isomorphic free groups share this geometric property.

Contribution

It demonstrates that homomorphisms between free groups with bi-invariant metrics are quasi-isometries only when they are isomorphisms, highlighting a rigidity property.

Findings

Free groups of different ranks are not bi-invariantly quasi-isometric.

Homomorphisms are quasi-isometries only if they are isomorphisms.

Bi-invariant word metrics distinguish free groups by rank.

Abstract

We prove that a homomorphism between free groups of finite rank equipped with the bi-invariant word metrics is a quasi-isometry if and only if it is an isomorphism.