The Euclidean Distortion of the Lamplighter Group

Abstract

We show that the cyclic lamplighter group C 2 ≀ C n embeds into Hilbert space with distortion \(\mathrm{O}(\sqrt{\log n})\) . This matches the lower bound proved by Lee et al. (Geom. Funct. Anal., 2009), answering a question posed in that paper. Thus, the Euclidean distortion of C 2 ≀ C n is \(\varTheta(\sqrt{\log n})\) . Our embedding is constructed explicitly in terms of the irreducible representations of the group. Since the optimal Euclidean embedding of a finite group can always be chosen to be equivariant, as shown by Aharoni et al. (Isr. J. Math. 52(3):251–265, 1985) and by Gromov (see de Cornulier et. al. in Geom. Funct. Anal., 2009), such representation-theoretic considerations suggest a general tool for obtaining upper and lower bounds on Euclidean embeddings of finite groups.