Coarse embeddings into a Hilbert space, Haagerup Property and Poincare inequalities

TL;DR

This paper establishes a characterization of when metric spaces and groups do not coarsely embed into Hilbert spaces, linking it to Poincaré inequalities, expanders, and properties like Haagerup and property T.

Contribution

It provides a new equivalence between coarse non-embeddability into Hilbert spaces and Poincaré inequalities, extending to group properties such as Haagerup and property T.

Findings

Metric spaces do not coarsely embed into Hilbert spaces iff they satisfy certain Poincaré inequalities.

Groups lack the Haagerup property iff they have relative property T with respect to certain probability measures.

Quantitative relations are established in terms of compression and unitary representations.

Abstract

We prove that a metric space does not coarsely embed into a Hilbert space if and only if it satisfies a sequence of Poincar\'e inequalities, which can be formulated in terms of (generalized) expanders. We also give quantitative statements, relative to the compression. In the equivariant context, our result says that a group does not have the Haagerup property if and only if it has relative property T with respect to a family of probabilities whose supports go to infinity. We give versions of this result both in terms of unitary representations, and in terms of affine isometric actions on Hilbert spaces.