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Article Dans Une Revue Geometric And Functional Analysis Année : 2007

Isometric group actions on hilbert spaces: Growth of cocycles

Alain Valette
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Résumé

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" Folner 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.

Dates et versions

hal-00364900 , version 1 (27-02-2009)

Identifiants

Citer

Yves de Cornulier, Romain Tessera, Alain Valette. Isometric group actions on hilbert spaces: Growth of cocycles. Geometric And Functional Analysis, 2007, 17 (3), pp.770-792. ⟨10.1007/s00039-007-0604-0⟩. ⟨hal-00364900⟩
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