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If one tries to embed a metric space uniformly in Hilbert space, how close to quasi-isometric could the embedding be? We answer this question for finite dimensional CAT(0) cube complexes and for hyperbolic groups. In particular, we show that the Hilbert space compression of any hyperbolic group is 1.

Abstract. Let G be either a non-elementary (word) hyperbolic group or a large group (both in the sense of Gromov). In this article we describe several approaches for constructing continuous families of periodic quotients of G with various properties.The first three methods work for any non-elementary hyperbolic group, producing three different continua of periodic quotients of G. They are based on the results and techniques, that were developed by Ivanov and Olshanskii in order to show that there exists an integer n such that G=G n is an infinite group of exponent n. The fourth approach starts with a large group G and produces a continuum of pairwise non-isomorphic periodic residually finite quotients.Speaking of a particular application, we use each of these methods to give a positive answer to a question of Wiegold from the Kourovka Notebook.