A characterization of superreflexivity through embeddings of lamplighter groups

Abstract

We prove that finite lamplighter groups $\{{\mathbb{Z}}_2\wr {\mathbb{Z}}_n{\}}_{n\ge 2}$ with a standard set of generators embed with uniformly bounded distortions into any non-superreflexive Banach space, and therefore form a set of test-spaces for superreflexivity. Our proof is inspired by the well known identification of Cayley graphs of infinite lamplighter groups with the horocyclic product of trees. We cover ${\mathbb{Z}}_2\wr {\mathbb{Z}}_n$ by three sets with a structure similar to a horocyclic product of trees, which enables us to construct well-controlled embeddings.

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Cited by in corpus (1)

• On the bi-Lipschitz geometry of lamplighter graphs