Uniform Embeddability into Hilbert Space

TL;DR

This paper characterizes when box spaces of residually finite groups can be uniformly embedded into Hilbert space, linking the absence of expander sequences to the Haagerup property.

Contribution

It establishes a necessary and sufficient condition for uniform embeddability of box spaces into Hilbert space based on expander sequences.

Findings

A box space contains no uniformly embedded expander sequence iff it embeds into Hilbert space.

Provides a sufficient condition for residually finite groups to have the Haagerup property.

Extends results to disjoint unions of Cayley graphs of finite groups with bounded degree.

Abstract

The open question of what prevents a metric space with bounded geometry from being uniformly embeddable in Hilbert space is answered here for box spaces of residually finite groups. We prove that a box space does not contain a uniformly embedded expander sequence if and only if it uniformly embeds in Hilbert space. In particular, this gives a sufficient condition for a residually finite group to have the Haagerup property. The main result holds in the more general setting of a disjoint union of Cayley graphs of finite groups with bounded degree.