  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.
