The euclidean distortion of the lamplighter Group

Abstract

We show that the cyclic lamplighter group C2{wreath product}Cn embeds into Hilbert space with distortion O(√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 C2{wreath product}Cn is Θ(√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. © Springer Science+Business Media, LLC 2009.