Box spaces of the free group that neither contain expanders nor embed into a Hilbert space

TL;DR

This paper constructs specific box spaces of a free group that neither contain coarsely embedded expanders nor embed into a Hilbert space, challenging previous assumptions about their relationship.

Contribution

It introduces a novel method of intersecting subgroup sequences to produce box spaces with unique embedding and expander properties.

Findings

Constructed box spaces that do not contain coarsely embedded expanders.

Demonstrated that certain box spaces do not embed into Hilbert space.

Proved that covering relations prevent coarsely embedded expanders in specific box spaces.

Abstract

We construct box spaces of a free group that do not coarsely embed into a Hilbert space, but do not contain coarsely embedded expanders. We do this by considering two sequences of subgroups of the free group: one which gives rise to a box space which forms an expander, and another which gives rise to a box space that can be coarsely embedded into a Hilbert space. We then take certain intersections of these subgroups, and prove that the corresponding box space contains generalized expanders. We show that there are no coarsely embedded expanders in the box space corresponding to our chosen sequence by proving that a box space that covers another box space of the same group that is coarsely embeddable into a Hilbert space cannot contain coarsely embedded expanders.