On coarse embeddings of amenable groups into hyperbolic graphs

TL;DR

This paper proves that finitely generated amenable groups can only embed into hyperbolic groups if they are virtually nilpotent, resolving a question about coarse embeddings and their limitations.

Contribution

It establishes a characterization of amenable groups that can embed into hyperbolic groups, showing they must be virtually nilpotent, and connects this to Lorentz geometry applications.

Findings

Amenable groups embedding into hyperbolic groups are virtually nilpotent.

Answer to Hume and Sisto's question on embeddings.

Application to Lorentz geometry by Charles Frances.

Abstract

In this note we prove that if a finitely generated amenable group admits a regular map to a direct product of a hyperbolic space and a euclidean space, then it must be virtually nilpotent. We deduce that an amenable group regularly embeds into a hyperbolic group if and only if it is virtually nilpotent, answering a question of Hume and Sisto. We describe an application to Lorentz geometry due to Charles Frances.