### Abstract

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 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 α*(G) := α_{L2}*(G) (respectively, α^{#}(G) := α_{L2}^{#}(G)). We show that if X has modulus of smoothness of power type p, then α_{X} ^{#}(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[d_{G}(W_{t},e)] ≥ ct^{β}, 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 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)}) = 1/2-2^{1-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 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 Enfio type.

Original language | English (US) |
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Article number | rnn076 |

Journal | International Mathematics Research Notices |

Volume | 2008 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2008 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*International Mathematics Research Notices*,

*2008*(1), [rnn076]. https://doi.org/10.1093/imrn/rnn076