The Euclidean Distortion of the Lamplighter Group

AbstractWe show that the cyclic lamplighter group C2≀Cn embeds into Hilbert space with distortion $\mathrm{O}(\sqrt{\log n})$ . 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≀Cn is $\varTheta(\sqrt{\log n})$ . 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 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.

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