A characterization of superreflexivity through embeddings of lamplighter groups

Ostrovskii, Mikhail; Randrianantoanina, Beata

doi:10.1090/proc/14526

A characterization of superreflexivity through embeddings of lamplighter groups
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by Mikhail I. Ostrovskii and Beata Randrianantoanina PDF

Proc. Amer. Math. Soc. 147 (2019), 4745-4755 Request permission

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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