Uniform Embeddability into Hilbert Space
Group Theory
2011-03-30 v2 Functional Analysis
Metric Geometry
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.
Cite
@article{arxiv.1101.0951,
title = {Uniform Embeddability into Hilbert Space},
author = {A. Khukhro},
journal= {arXiv preprint arXiv:1101.0951},
year = {2011}
}
Comments
This paper has been withdrawn by the author due to an error in the proof of the main theorem