The Euclidean distortion of the lamplighter group

TL;DR

This paper establishes the precise Euclidean distortion of the lamplighter group, showing it grows as the square root of the logarithm of the group's size, and provides an explicit embedding using group representations.

Contribution

It proves the exact order of Euclidean distortion for the lamplighter group and constructs an explicit embedding based on irreducible representations.

Findings

Euclidean distortion of the lamplighter group is Θ(√log n)

Constructs explicit embeddings using irreducible representations

Answers a previously open question about the group's embedding distortion

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.