Embeddable box spaces of free groups

TL;DR

This paper constructs new examples of metric spaces derived from free groups that coarsely embed into Hilbert space without possessing Yu's property A, expanding understanding of coarse embeddings.

Contribution

It generalizes the construction of box spaces for free groups using the derived m-series, providing new examples of spaces with coarse embeddings into Hilbert space.

Findings

Box spaces of free groups with derived m-series coarsely embed into Hilbert space.

These spaces do not have Yu's property A.

The construction broadens the class of known coarse embeddable metric spaces.

Abstract

We generalize the construction of Arzhantseva, Guentner and Spakula of a box space of the free group which admits a coarse embedding into Hilbert space. We show that for a finitely generated free group, the box space corresponding to the derived $m$-series (for any integer $m\geq 2$) coarsely embeds into Hilbert space. This gives new examples of metric spaces with bounded geometry which coarsely embed into Hilbert space but do not have Yu's property A.