A characterization of superreflexivity through embeddings of lamplighter groups

Abstract

We prove that finite lamplighter groups { Z 2 ≀ Z n } n ≥ 2 \{\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 Z 2 ≀ Z n \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.