A characterization of superreflexivity through embeddings of lamplighter groups

We prove that finite lamplighter groups {Z(2) (sic) Z(n)}(n >= 2) with a standard set of generators embed with uniformly bounded distortions into any non-superreflexive B-n-ch sp-ce -nd therefore form - set of test sp-ces for superreflexivity. Our proof is inspired by the well- known identific-tion of C-yley gr-phs of infinite l-mplighter groups with the horocyclic product of trees. We coverZ(2) (sic) Z(n) by three sets with - structure simil-r to - horocyclic product of trees, which en-bles us to construct well- controlled embeddings.