A quasi-isometric embedding theorem for groups

Abstract

We show that every group H of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group G such that G is amenable (respectively, solvable, satisfies a non-trivial identity, elementary amenable, of finite decomposition complexity, etc.) whenever H is. We also discuss some applications to compression functions of Lipschitz embeddings into uniformly convex Banach spaces, Følner functions, and elementary classes of amenable groups. arXiv:1202.6437v4 [math.GR] 30 Apr 2013In particular, our embedding allows us to carry over a wide range of properties from the group H to the group G. For instance, we have the following.Corollary 1.2. In the notation of Theorem 1.1, if H is solvable (respectively, satisfies a non-trivial identity, elementary amenable, amenable, has property A, has finite decomposition complexity, uniformly embeds in a Hilbert space, etc.), then so is G.Recall that property A was introduced by Yu [30] and groups of finite decomposition complexity were introduced by Guentner, Tessera, and Yu [16] with motivation coming from the Novikov conjecture and topological rigidity of manifolds, respectively. For definitions, properties, and applications we refer to [16,25,30] and references therein. Groups uniformly embeddable in Hilbert spaces are discussed below.The classes of elementary amenable groups, amenable groups, countable groups with property A, countable groups of finite decomposition complexity, and countable groups uniformly embeddable in a Hilbert space contain abelian groups and are closed with respect to

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