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

2007/06/13 by Austin, Tim, Naor, Assaf, Peres, Yuval
#FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.0706.1943

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→ L₂ and a constant c>0 such that for all x,y∈ G we have ‖f(x)-f(y)‖₂≥ cd(x,y)^α. In \citeAGS06 it was shown that the Hilbert compression exponent of the wreath product \Z\bwr \Z is at most \frac34, and in \citeNP07 was proved that this exponent is at least \frac23. Here we show that \frac23 is the correct value. Our proof is based on an application of K. Ball's notion of Markov type.

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