The wreath product of ℤ with ℤ has Hilbert compression exponent 2/3

Tim Austin, Assaf Naor, Yuval Peres

Research output: Contribution to journalArticle

Abstract

Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all α ≥ 0 such that there exists a Lipschitz mapping f : G → L 2 and a constant c > 0 such that for all x,y ∈ G we have ||f(x) - f(y)|| 2 ≥ cd(x,y) α. It was previously known that the Hilbert compression exponent of the wreath product ℤ ∼ ℤ is between 2/3 and 3/4. Here we show that 2/3 is the correct value. Our proof is based on an application of K. Ball's notion of Markov type.

Original languageEnglish (US)
Pages (from-to)85-90
Number of pages6
JournalProceedings of the American Mathematical Society
Volume137
Issue number1
DOIs
StatePublished - Jan 2009

Fingerprint

Wreath Product
Hilbert
Compression
Exponent
Lipschitz Mapping
Finitely Generated Group
Supremum
Finite Set
Ball
Generator
Metric

Keywords

  • Coarse geometry
  • Geometric group theory
  • Hilbert compression exponents
  • Markov type

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

The wreath product of ℤ with ℤ has Hilbert compression exponent 2/3. / Austin, Tim; Naor, Assaf; Peres, Yuval.

In: Proceedings of the American Mathematical Society, Vol. 137, No. 1, 01.2009, p. 85-90.

Research output: Contribution to journalArticle

Austin, Tim ; Naor, Assaf ; Peres, Yuval. / The wreath product of ℤ with ℤ has Hilbert compression exponent 2/3. In: Proceedings of the American Mathematical Society. 2009 ; Vol. 137, No. 1. pp. 85-90.
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