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Isometric Group Actions on Hilbert Spaces: Growth of Cocycles

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

In: GAFA Geometric And Functional Analysis, 2007, vol. 17, no. 3, p. 770-792

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    Titre
    • Isometric Group Actions on Hilbert Spaces: Growth of Cocycles
    Auteur
    • de Cornulier, Yves. IRMAR, Campus de Beaulieu, F-35042, Rennes Cedex, France
    • Tessera, Romain. Department of Mathematics, Vanderbilt University, Stevenson Center, 37240, Nashville, TN, USA
    • Valette, Alain. Institut de Mathématiques, Université de Neuchâtel, rue Emile Argand 11, CH-2007, Neuchâtel, Switzerland
    Type de document
    • Postprint
    Langue
    • Anglais
    Publié dans
    • GAFA Geometric And Functional Analysis, 2007, vol. 17, no. 3, p. 770-792. Birkhäuser-Verlag; www.birkhäuser.ch
    Autre version électronique
    Identifiant OAI-PMH
    • oai:doc.rero.ch:321643
    Summary
    Abstract.: We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing polycyclic groups and connected amenable Lie groups: either G has no quasi-isometric embedding into a Hilbert space, or G admits a proper cocompact action on some Euclidean space. On the other hand, noting that almost coboundaries (i.e. 1-cocycles approximable by bounded 1-cocycles) have sublinear growth, we discuss the converse, which turns out to hold for amenable groups with "controlled” Følner sequences; for general amenable groups we prove the weaker result that 1-cocycles with sufficiently small growth are almost coboundaries. Besides, we show that there exist, on a-T-menable groups, proper cocycles with arbitrary small growth