Coarse non-amenability and coarse embeddings

TL;DR

This paper constructs the first example of a metric space that is coarsely non-amenable yet can still be embedded into a Hilbert space, challenging previous assumptions about their relationship.

Contribution

It provides the first known example of a coarsely non-amenable space with bounded geometry that admits a coarse embedding into a Hilbert space.

Findings

First example of coarsely non-amenable space with property A

Demonstrates such a space can embed into Hilbert space

Challenges previous beliefs about non-amenability and embeddings

Abstract

We construct the first example of a coarsely non-amenable (= without Guoliang Yu's property A) metric space with bounded geometry which coarsely embeds into a Hilbert space.