Embeddings of discrete groups and the speed of random walks

Details Embeddings of discrete groups and the speed of random walks Embeddings of discrete groups and the speed of random walks

2007-08-07

arxiv.org/abs/0708.0853v3

Mathematics

Metric Geometry

24 pages. Added Remark 6.4 and made minor changes in new version

Scientific paper

For a finitely generated group G and a banach space X let \alpha^*_X(G) (respectively \alpha^#_X(G)) be the supremum over all \alpha\ge 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f:G\to X and c>0 such that for all x,y\in G we have \|f(x)-f(y)\|\ge c\cdot d_G(x,y)^\alpha. In particular, the Hilbert compression exponent (respectively the equivariant Hilbert compression exponent) of G is \alpha^*(G)=\alpha^*_{L_2}(G) (respectively \alpha^#(G)= \alpha_{L_2}^#(G)). We show that if X has modulus of smoothness of power type p, then \alpha^#_X(G)\le \frac{1}{p\beta^*(G)}. Here \beta^*(G) is the largest \beta\ge 0 for which there exists a set of generators S of G and c>0 such that for all t\in \N we have \E\big[d_G(W_t,e)\big]\ge ct^\beta, where \{W_t\}_{t=0}^\infty is the canonical simple random walk on the Cayley graph of G determined by S, starting at the identity element. This result is sharp when X=L_p, generalizes a theorem of Guentner and Kaminker and answers a question posed by Tessera. We also show that if \alpha^*(G)\ge 1/2 then \alpha^*(G\bwr \Z)\ge \frac{2\alpha^*(G)}{2\alpha^*(G)+1}. This improves the previous bound due to Stalder and ValetteWe deduce that if we write \Z_{(1)}= \Z and \Z_{(k+1)}\coloneqq \Z_{(k)}\bwr \Z then \alpha^*(\Z_{(k)})=\frac{1}{2-2^{1-k}}, and use this result to answer a question posed by Tessera in on the relation between the Hilbert compression exponent and the isoperimetric profile of the balls in G. We also show that the cyclic lamplighter groups C_2\bwr C_n embed into L_1 with uniformly bounded distortion, answering a question posed by Lee, Naor and Peres. Finally, we use these results to show that edge Markov type need not imply Enflo type.

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