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We show that every metric space with bounded geometry uniformly embeds into a direct sum of l p (N) spaces (p's going off to infinity). In particular, every sequence of expanding graphs uniformly embeds into such a reflexive Banach space even though no such sequence uniformly embeds into a fixed l p (N) space. In the case of discrete groups we prove the analogue of a-T-menability - the existence of a metrically proper affine isometric action on a direct sum of l p (N) spaces.

Uniform embeddings of bounded geometry spaces into reflexive Banach space