The Euclidean Distortion of the Lamplighter Group
We show that the cyclic lamplighter group C(2) (sic) C(n) embeds into Hilbert space with distortion O(root 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) (sic) C(n) is Theta(root 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.
lamplighter group bi-lipschitz distortion