Exactness and Uniform Embeddability of Discrete Groups

Guentner, Erik; Kaminker, Jerome

doi:10.1112/s0024610704005897

Cited by 80 publications

(123 citation statements)

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“…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…”

Section: Introductionsupporting

confidence: 54%

“…Since for a nonamenable group G we have β * (G) = 1 (see [25,43]), Theorem 1.1 implies the Guentner-Kaminker theorem, while generalizing it to non-Hilbertian targets (when the target space X is a Hilbert space our method yields a very simple new proof of the Guentner-Kaminker theorem-see Remark 2.6). Note that both known proofs of the Guentner-Kaminker theorem, namely the original proof in [20] and the new proof discovered by de Cornulier, Tessera and Valette in [14], rely crucially on the fact that X is a Hilbert space. It follows in particular from Theorem 1.1 that for 2 ≤ p < ∞, if α # p (G) > 1 2 then G is amenable.…”

Section: Theorem 11 Let X Be a Banach Space Which Has Modulus Of Smmentioning

confidence: 99%

“…A theorem of Guentner and Kaminker [20] states that if α # (G) > 1 2 then G is amenable. Since for a nonamenable group G we have β * (G) = 1 (see [25,43]), Theorem 1.1 implies the Guentner-Kaminker theorem, while generalizing it to non-Hilbertian targets (when the target space X is a Hilbert space our method yields a very simple new proof of the Guentner-Kaminker theorem-see Remark 2.6).…”

Section: Theorem 11 Let X Be a Banach Space Which Has Modulus Of Smmentioning

confidence: 99%

“…For p ≥ 1 we write α * L p (G) = α * p (G) and when p = 2 we write α * 2 (G) = α * (G). The parameter α * (G) is called the Hilbert compression exponent of G. This quasi-isometric group invariant was introduced by Guentner and Kaminker in [20]. We refer to the papers [20,11,3,14,37,2,36,13] and the references therein for background on this topic and several interesting applications.…”

Section: Introductionmentioning

confidence: 99%

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Embeddings of Discrete Groups and the Speed of Random Walks

Naor

Peres

2008

International Mathematics Research Notices

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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 (We show that if X has modulus of smoothness of power type p,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 ∈ N 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. This improves the previous bound due to Stalder and Valette [36]. We deduce that if we write2−2 1−k , 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.

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“…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…”

Section: Introductionsupporting

confidence: 54%

Section: Theorem 11 Let X Be a Banach Space Which Has Modulus Of Smmentioning

confidence: 99%

Section: Theorem 11 Let X Be a Banach Space Which Has Modulus Of Smmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Embeddings of Discrete Groups and the Speed of Random Walks

Naor

Peres

2008

International Mathematics Research Notices

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“…Let α Y (X) be the compression exponent of X in Y (Y-compression of X in short) introduced by Guentner and Kaminker [13]. In other words, α Y (X) is the supremum of all numbers ≤ α ≤ so that there exist f : X → Y, τ ∈ ( , ∞), and C := C(τ) ∈ [ , ∞) such that if d X (x, y) ∈ [τ, ∞) then…”

Section: Proposition 54 (I) If a Metric Space X Is Nearly Isometricmentioning

confidence: 99%

Tight Embeddability of Proper and Stable Metric Spaces

Baudier

Lancien

2015

Analysis and Geometry in Metric Spaces

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Abstract:We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for p ∈ [ , ∞], every proper subset of Lp is almost Lipschitzly embeddable into a Banach space X if and only if X contains uniformly the n p 's. We also sharpen a result of N. Kalton by showing that every stable metric space is nearly isometrically embeddable in the class of re exive Banach spaces.

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