TY - JOUR
T1 - Embeddings of discrete groups and the speed of random walks
AU - Naor, Assaf
AU - Peres, Yuval
PY - 2008
Y1 - 2008
N2 - Let G be a group generated by a finite set S and equipped with the associated leftinvariant word metric dG. For a Banach space X, let αX*(G) (respectively, αX#(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 α*(G) := αL2*(G) (respectively, α#(G) := αL2#(G)). We show that if X has modulus of smoothness of power type p, then αX #(G) ≤ 1/pβ*(G). 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 t ∈ we have E[dG(Wt,e)] ≥ ctβ, where {Wt}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 = Lp, generalizes a theorem of Guentner and Kaminker [20], and answers a question posed by Tessera [37]. We also show that, if α*(G) ≥ 1/2 then α*(G ) ≥ 2α*(G)/2α*(G)+1. This improves the previous bound due to Stalder and Valette [36]. We deduce that if we write (1) := and (k+1) := (k) then α*((k)) = 1/2-21-k, and use this result to answer a question posed by Tessera in [37] on the relation between the Hubert compression exponent and the isoperimetric profile of the balls in G. We also show that the cyclic lamplighter groups C2 Cn embed into L1 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 Enfio type.
AB - Let G be a group generated by a finite set S and equipped with the associated leftinvariant word metric dG. For a Banach space X, let αX*(G) (respectively, αX#(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 α*(G) := αL2*(G) (respectively, α#(G) := αL2#(G)). We show that if X has modulus of smoothness of power type p, then αX #(G) ≤ 1/pβ*(G). 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 t ∈ we have E[dG(Wt,e)] ≥ ctβ, where {Wt}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 = Lp, generalizes a theorem of Guentner and Kaminker [20], and answers a question posed by Tessera [37]. We also show that, if α*(G) ≥ 1/2 then α*(G ) ≥ 2α*(G)/2α*(G)+1. This improves the previous bound due to Stalder and Valette [36]. We deduce that if we write (1) := and (k+1) := (k) then α*((k)) = 1/2-21-k, and use this result to answer a question posed by Tessera in [37] on the relation between the Hubert compression exponent and the isoperimetric profile of the balls in G. We also show that the cyclic lamplighter groups C2 Cn embed into L1 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 Enfio type.
UR - http://www.scopus.com/inward/record.url?scp=77952071489&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=77952071489&partnerID=8YFLogxK
U2 - 10.1093/imrn/rnn076
DO - 10.1093/imrn/rnn076
M3 - Article
AN - SCOPUS:77952071489
SN - 1073-7928
VL - 2008
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 1
M1 - rnn076
ER -