Isometric group actions on Hilbert spaces: growth of cocycles

Details Isometric group actions on Hilbert spaces: growth of cocycles Isometric group actions on Hilbert spaces: growth of cocycles

2005-09-22

arxiv.org/abs/math/0509527v2

Geom. Funct. Anal. 17(3) (2007), 770-792

Mathematics

Group Theory

26 pages, no figure. To appear in Geom. Funct. Anal

Scientific paper

10.1007/s00039-007-0604-0

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 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.

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