Embeddings of Discrete Groups and the Speed of Random Walks

Let G be a group generated by a finite set S and equipped with the associated left-invariant word metric d G . For a Banach space X, let α * X (G) (respectively, α / # X (G)) be the supremum over all α ≥ 0 such that there exists a Lipschitz mapping (respectively, an equivariant mapping) f:G → X and c > 0 such that for all x, y ∈ G we have ‖ f(x) − f(y)‖ ≥ c · d G (x, y) α . In particular, the Hilbert compression exponent (respectively, the equivariant Hilbert compression exponent) of G is (respectively, ). We show that if X has modulus of smoothness of power type p, then . Here β * (G) is the largest β ≥ 0 for which there exists a set of generators S of G and c > 0, such that for all we have , where { W t } ∞ t=0 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 (20), and answers a question posed by Tessera (37). We also show that, if then . This improves the previous bound due to Stalder and Valette (36). We deduce that if we write and then and use this result to answer a question posed by Tessera in (37) 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 ≀ C n embed into L 1 with uniformly bounded distortion, answering a question posed by Lee, Naor, and Peres in (26). Finally, we use these results to show that edge Markov type need not imply Enflo type.