The Euclidean distortion of the lamplighter group

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dc.creatorAustin, Tim
dc.creatorNaor, Assaf
dc.creatorValette, Alain
dc.date2007-05-31
dc.date.accessioned2026-07-07T08:05:10Z
dc.date.available2026-07-07T08:05:10Z
dc.descriptionWe 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 $Θ(\sqrt{\log n})$. 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.
dc.identifierhttps://arxiv.org/abs/0705.4662
dc.identifierhttp://arxiv.org/abs/0705.4662
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/130191
dc.subjectMetric Geometry
dc.subjectFunctional Analysis
dc.subjectGroup Theory
dc.subject46B20, 54E40, 52C99
dc.titleThe Euclidean distortion of the lamplighter group
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