Isometric Group Actions on Hilbert Spaces: Growth of Cocycles

Cornulier, Yves de; Tessera, Romain; Valette, Alain

doi:10.1007/s00039-007-0604-0

Cited by 73 publications

(106 citation statements)

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“…In the Hilbertian case, when G is amenable we have α * (G) = α # (G). This was proved by by Aharoni, Maurey and Mityagin [1] (see also Chapter 8 in [9]) when G is Abelian, and by Gromov for general amenable groups (see [14]). …”

Section: Introductionmentioning

confidence: 83%

“…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%

“…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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Section: Introductionmentioning

confidence: 83%

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

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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“…One thus concludes that, for a generic g ∈ D ∞ , the sequence ( c(g n ) ) is unbounded, but it has the zero vector as an accumulation point. Furthermore, the orbits of the corresponding isometry A(g) = Ψ(g) + c(g) are unbounded and recurrent, in the sense that all of their points are accumulation points (see [59] for more on unbounded, recurrent actions by isometries on Hilbert spaces). Exercise 5.2.28.…”

Section: The Statement Of the Resultsmentioning

confidence: 99%

Groups of Circle Diffeomorphisms

Navas¹

2011

150

171

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The original version (in Spanish) of this text was prepared for a mini-course in Antofagasta, Chile. Subsequently, enlarged and revised versions were published in the series Monografías del IMCA (Perú) and Ensaios Matemáticos (Brasil). This translation arose from the necessity of making this text accessible to a larger audience. I would like to thank Juan Rivera-Letelier for his invitation to the II Workshop on Dynamical Systems ( 2001), for which the original version was prepared, and Roger Metzger for his invitation to IMCA (2006), where part of this material was presented. I would also like to thank both Étienne Ghys and Maria Eulália Vares for motivating me and allowing me to publish the revised Spanish version in Brasil, as well as all the people who strongly encouraged me to conclude this English version.This work was partially supported by CONICYT and PBCT via the Research Network on Low Dimensional Dynamics. I would also to acknowledge the support of both the UMPA Department of the École Normale Supérieure de Lyon, where the idea of writing this text was born during my PhD thesis, and the Institut des Hautes Études Scientifiques, where the original notes started taking its definite form while I benefited from a one-year postdoctoral position.This work owes a lot to many of my colleges. Several remarks spread throughout the text and the content of some examples and exercises were born during fruitful and quite stimulating discussions. It is then a pleasure to thank Sylvain Crovisier (Proposition 4.2.25),

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“…For instance, Shalom shows that an amenable finitely generated group with Property H F D has a finite index subgroup with infinite abelianization [25,Theorem 4.3.1]. In [6], we prove [6,Theorem 4.3] that an amenable finitely generated group with Property H F D cannot quasi-isometrically embed into a Hilbert space unless it is virtually abelian. It is interesting and natural to extend the definition of Property H F D to isometric representations of groups on certain classes of Banach spaces.…”

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confidence: 91%

Vanishing of the first reduced cohomology with values in an L^p-representation

Tessera¹

2009

Ann. inst. Fourier

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Abstract. -We prove that the first reduced cohomology with values in a mixing L p -representation, 1 < p < ∞, vanishes for a class of amenable groups including connected amenable Lie groups. In particular this solves for this class of amenable groups a conjecture of Gromov saying that every finitely generated amenable group has no first reduced p -cohomology. As a byproduct, we prove a conjecture by Pansu. Namely, the first reduced L p -cohomology on homogeneous, closed at infinity, Riemannian manifolds vanishes. We also prove that a Gromov hyperbolic geodesic metric measure space with bounded geometry admitting a bi-Lipschitz embedded 3-regular tree has non-trivial first reduced L p -cohomology for large enough p. Combining our results with those of Pansu, we characterize Gromov hyperbolic homogeneous manifolds: these are the ones having non-zero first reduced L p -cohomology for some 1 < p < ∞.Résumé. -Nous prouvons que pour une classe de groupes moyennables, incluant tous les groupes moyennables de Lie connexes, la cohomologie réduite en degré 1 à valeurs dans une représentation mélangeante sur un espace L p , pour p > 1, est nulle. En particulier, cela démontre pour cette classe de groupes moyennables une conjecture de Gromov s'appliquant à tous les groupes de type fini moyennables. Nous obtenons également la version "de Lie" de cette conjecture, qui avait été formulée par Pansu. Nous montrons par ailleurs qu'un espace métrique hyperbolique possédant un arbre 3-regulier quasi-isométriquement plongé a un premier groupe de cohomologie L p réduite non trivial pour p assez grand. Finalement, en combinant nos résultats avec ceux de Pansu, nous obtenons une caractérisation des variétés riemanniennes homogènes hyperboliques au sens de Gromov : ce sont celles qui possèdent de la cohomologie L p réduite en degré 1 pour p assez grand.

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Isometric Group Actions on Hilbert Spaces: Growth of Cocycles

Cited by 73 publications

References 22 publications

Embeddings of Discrete Groups and the Speed of Random Walks

Embeddings of Discrete Groups and the Speed of Random Walks

Groups of Circle Diffeomorphisms

Vanishing of the first reduced cohomology with values in an L^p-representation

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