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Abstract: We prove that a metric space does not coarsely embed into a Hilbert space ifand only if it satisfies a sequence of Poincar\-e inequalities, which can beformulated in terms of generalized expanders. We also give quantitativestatements, relative to the compression. In the equivariant context, our resultsays that a group does not have the Haagerup property if and only if it hasrelative property T with respect to a family of probabilities whose supports goto infinity. We give versions of this result both in terms of unitaryrepresentations, and in terms of affine isometric actions on Hilbert spaces.



Author: Romain Tessera

Source: https://arxiv.org/



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