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

Source: [Are free groups of different ranks bi-invariantly quasi-isometric?](<https://www.sota2.com/research/paper/68fa5cbc1e209131ce9d3ad1>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T17:49:20.283Z

Authors: Jarek K\\k{e}dra, Assaf Libman

Published: 2024-07-25T13:24:38.000Z

## 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.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2407.18027>)

[PDF](<https://arxiv.org/pdf/2407.18027.pdf>)