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

Tim Austin, Assaf Naor, Yuval Peres

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


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
Issue number1
StatePublished - Jan 2009
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


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


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