Box spaces, group extensions and coarse embeddings into Hilbert space

TL;DR

This paper studies how the coarse embeddability of box spaces into Hilbert space is affected by group extensions, providing new examples of spaces that embed into Hilbert space without property A.

Contribution

It proves that certain group extensions produce box spaces that coarsely embed into Hilbert space, expanding known classes of such metric spaces.

Findings

Semidirect products of free groups and residually finite amenable groups have coarsely embeddable box spaces.

Provides new examples of metric spaces with bounded geometry that embed into Hilbert space but lack property A.

Generalizes previous examples by Arzhantseva, Guentner, and Spakula.

Abstract

We investigate how coarse embeddability of box spaces into Hilbert space behaves under group extensions. In particular, we prove a result which implies that a semidirect product of a finitely generated free group by a finitely generated residually finite amenable group has a box space which coarsely embeds into Hilbert space. This provides a new class of examples of metric spaces with bounded geometry which coarsely embed into Hilbert space but do not have property A, generalising the example of Arzhantseva, Guentner and Spakula.