Property (T) and rigidity for actions on Banach spaces

Bader, Uri; Furman, Alex; Gelander, Tsachik; Monod, Nicolas

doi:10.1007/s11511-007-0013-0

Cited by 137 publications

(350 citation statements)

References 51 publications

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“…The above corollary about the extension of group actions was previously noted in [6] under the additional restriction that 1 < p 2Z, as a simple corollary of an abstract extension theorem due to Hardin [22] (alternatively this is also a corollary of the classical Plotkin-Rudin theorem [33,35] We shall now pass to the proof of Theorem 2.1. We will use uniform smoothness via the following famous inequality due to Pisier [31] (for the explicit constant below see Theorem 4.2 in [28]).…”

Section: Standard Complex Gaussian Random Variables) This Fact Also mentioning

confidence: 83%

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: Standard Complex Gaussian Random Variables) This Fact Also mentioning

confidence: 83%

Embeddings of Discrete Groups and the Speed of Random Walks

Naor

Peres

2008

International Mathematics Research Notices

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“…This theorem has been generalized in [5] to isometric actions on L p spaces. This allows obtaining C 1+τ versions of the preceding results for every τ > 0.…”

Section: Relative Property (T) and Haagerup's Propertymentioning

confidence: 99%

Groups of Circle Diffeomorphisms

Navas¹

2011

153

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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“…A basic example of a measure equivalent pair (Γ, Λ) is a pair of two lattices in a common locally compact second countable group. Moreover, if these two in the example above are irreducible higher rank lattices, then this measure equivalence is known to satisfy the L p -integrability condition for all p ∈ [1, ∞), see for instance [BFGM07,Section 8]. Therefore, item (ii) of Theorem 1.1 generalizes the FarbKaimanovich-Masur and the Bridson-Wade superrigidity of higher rank lattices, provided that the corresponding higher rank algebraic group has no rank one factor, to groups possibly with no arithmetic backgrounds.…”

Section: (I) For Every Acylindrically Hyperbolic Group G Every Groupmentioning

confidence: 99%

Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles

Mimura

2017

J. Eur. Math. Soc.

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Abstract. We prove that every homomorphism from the elementary Chevalley group over a finitely generated unital commutative ring associated with a reduced irreducible classical root system of rank at least 2, and ME analogues of such groups, into acylindrically hyperbolic groups has an absolutely elliptic image. This result provides a non-arithmetic generalization of homomorphism superrigidity of Farb-Kaimanovich-Masur and Bridson-Wade.

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Property (T) and rigidity for actions on Banach spaces

Cited by 137 publications

References 51 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

Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles

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