The Euclidean Distortion of the Lamplighter Group

We show that the cyclic lamplighter group C2 Cn embeds into Hilbert space with distortion O log n . This matches the lower bound proved by Lee, Naor and Peres in [14], answering a question posed in that paper. Thus the Euclidean distortion of C2 Cn is 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 [1] and by Gromov (see [9]), such representation-theoretic considerations suggest a general tool for obtaining upper and lower bounds on Euclidean embeddings of finite groups.

• Type: Journal
• Year: 2010
• Where: DCG
• Authors: Tim Austin, Assaf Naor, Alain Valette
• Keywords: DCG 2010, Euclidean Distortion, Finite Groups, Group C2 Cn