Uniform Embeddability into Hilbert Space

Details Uniform Embeddability into Hilbert Space Uniform Embeddability into Hilbert Space

2011-01-05

arxiv.org/abs/1101.0951v2

Mathematics

Group Theory

This paper has been withdrawn by the author due to an error in the proof of the main theorem

Scientific paper

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.

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