The euclidean distortion of the lamplighter Group

Tim Austin, Assaf Naor, Alain Valette

Research output: Contribution to journalArticle

Abstract

We show that the cyclic lamplighter group C2{wreath product}Cn embeds into Hilbert space with distortion O(√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{wreath product}Cn is Θ(√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.

Original languageEnglish (US)
Pages (from-to)55-74
Number of pages20
JournalDiscrete and Computational Geometry
Volume44
Issue number1
DOIs
StatePublished - Jul 2010

Fingerprint

Euclidean
Wreath Product
Hilbert spaces
Finite Group
Cyclic group
Irreducible Representation
Equivariant
Upper and Lower Bounds
Hilbert space
Lower bound

Keywords

  • Bi-Lipschitz distortion
  • Lamplighter group

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Discrete Mathematics and Combinatorics
  • Geometry and Topology

Cite this

The euclidean distortion of the lamplighter Group. / Austin, Tim; Naor, Assaf; Valette, Alain.

In: Discrete and Computational Geometry, Vol. 44, No. 1, 07.2010, p. 55-74.

Research output: Contribution to journalArticle

Austin, Tim ; Naor, Assaf ; Valette, Alain. / The euclidean distortion of the lamplighter Group. In: Discrete and Computational Geometry. 2010 ; Vol. 44, No. 1. pp. 55-74.
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