[Banach] Abstract of a paper by Tim Austin, Assaf Naor, and Yuval Peres

This is an announcement for the paper "The wreath product of Z with Z has Hilbert compression exponent 2/3" by Tim Austin, Assaf Naor, and Yuval Peres.

Abstract: We consider the wreath product $\Z\bwr \Z $, and prove that any Lipschitz function $f:\Z\bwr \Z \to L_2$ satisfies $$\liminf_{d_{\Z\bwr\Z}(x,y)\to \infty}\frac{|f(x)-f(y)|_2}{d_{\Z\bwr\Z}(x,y)^{2/3}}<\infty. $$ On the other hand, as as shown by Tessera in~\cite{Tess06}, there exists a Lipschitz function $g:\Z\bwr \Z \to L_2$ and a real $c>0$ such that $|f(x)-f(y)|_2 \ge c,d_{\Z\bwr\Z}(x,y)^{2/3}$ for all $x,y \in \Z\bwr\Z$. Thus the Hilbert compression exponent of $\Z\bwr \Z$ is exactly $\frac23$, answering a question posed by Arzhantseva, Guba and Sapir~\cite{AGS06} and by Tessara~\cite{Tess06}. Our proof is based on an application of K. Ball's notion of Markov type.

Archive classification: math.MG math.FA

The source file(s), ZwreathZ.bbl: 3412 bytes

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/0706.1943

or

http://arXiv.org/abs/0706.1943

or by email in unzipped form by transmitting an empty message with subject line

uget 0706.1943

or in gzipped form by using subject line

get 0706.1943

to: math@arXiv.org.