[Banach] Abstract of a paper by Tim Austin, Assaf Naor, and Alain Valette

This is an announcement for the paper "The Euclidean distortion of the lamplighter group" by Tim Austin, Assaf Naor, and Alain Valette.

Abstract: We show that the cyclic lamplighter group $C_2 \bwr C_n$ embeds into Hilbert space with distortion ${\rm O}\left(\sqrt{\log n}\right)$. 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 $\Theta\left(\sqrt{\log n}\right)$. 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.

Archive classification: math.MG math.FA

Mathematics Subject Classification: 46B20, 54E40, 52C99

The source file(s), LAMP-official.bbl: 3624 bytes

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