A characterization of superreflexivity through embeddings of lamplighter groups

Abstract

We prove that finite lamplighter groups {Z 2 ≀ Z n } n≥2 with a standard set of generators embed with uniformly bounded distortions into any nonsuperreflexive 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 by three sets with a structure similar to a horocyclic product of trees, which enables us to construct well-controlled embeddings.

Stochastic approximation of lamplighter metrics

“…It was shown in [49] that finite cyclic lamplighter groups bi‐Lipschitzly embed into $$L_1$$ with distortion bounded above independently of the size of the cyclic group. It follows from this result and general principles that the lamplighter group over the infinite cyclic group bi‐Lipschitzly embeds into $$L_1$$ (an alternative direct proof is also given in [49] and we refer to [54] for an argument using a description of this lamplighter group in terms of a horocyclic product of trees [66]). These results were extended to non‐superreflexive Banach space targets in [54].…”

“…It follows from this result and general principles that the lamplighter group over the infinite cyclic group bi‐Lipschitzly embeds into $$L_1$$ (an alternative direct proof is also given in [49] and we refer to [54] for an argument using a description of this lamplighter group in terms of a horocyclic product of trees [66]). These results were extended to non‐superreflexive Banach space targets in [54]. That the lamplighter group over a finitely generated free group bi‐Lipschitzly embeds into $$L_1$$, is a by‐product of the work of de Cornulier, Stadler and Valette [17] on the equivariant $$L_1$$‐compression of wreath products.…”