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Uniform embeddability of relatively hyperbolic groups

Abstract (from the preprint)

Let G be a finitely generated group which is hyperbolic relative to a finite family A, B, ..., C of subgroups. We prove that G is uniformly embeddable in a Hilbert space if and only if each subgroup is uniformly embeddable in a Hilbert space. A 'gluing' technique for proving uniform embeddability of metric spaces is introduced, and plays a fundamental role in the proof of the result. Further applications of the gluing technique will be considered elsewhere.