Embeddings of discrete groups and the speed of random walks

Abstract

For a finitely generated group G and a banach space X let α^*_X(G) (respectively α^#_X(G)) be the supremum over all α\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)^α. In particular, the Hilbert compression exponent (respectively the equivariant Hilbert compression exponent) of G is α^*(G)=α^*_{L_2}(G) (respectively α^#(G)= α_{L_2}^#(G)). We show that if X has modulus of smoothness of power type p, then α^#_X(G)\le \frac{1}{pβ^*(G)}. Here β^*(G) is the largest β\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^β, 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 α^*(G)\ge 1/2 then α^*(G\bwr \Z)\ge \frac{2α^*(G)}{2α^*(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 α^*(\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.