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

2008/02/19 by Romain Tessera, Tessera, Romain
Mathematics · #43A85. #51F99 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:43A85. #msc:51F99

paper · pdf · doi:10.48550/arxiv.0802.2541

14 pages

openalex publication_date 2008/02/19 · arxiv created 2008/03/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that a metric space does not coarsely embed into a Hilbert space if and only if it satisfies a sequence of Poincaré 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.

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