The Euclidean distortion of the lamplighter group

Abstract

We show that the cyclic lamplighter group $C_2 \bwr C_n$ embeds into Hilbert space with distortion ${\rm O}(\sqrt{\log n})$. This matches the lower bound proved by Lee, Naor and Peres in \cite{LeeNaoPer}, answering a question posed in that paper. Thus the Euclidean distortion of $C_2 \bwr C_n$ is $\Theta(\sqrt{\log n})$. 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, Maurey and Mityagin \cite{AhaMauMit} and by Gromov (see \cite{deCTesVal}), such representation-theoretic considerations suggest a general tool for obtaining upper and lower bounds on Euclidean embeddings of finite groups.