## Isometric group actions on Hilbert spaces: growth of cocycles.(English)Zbl 1129.22004

Let $$G$$ be a locally compact group, and $$\alpha$$ an affine isometric action on an affine Hilbert space $$\mathcal H$$. The function $$b: G\mapsto {\mathcal H}$$ defined by $$b(g)=\alpha(g)(0)$$ is called a $$1$$-cocycle, and the function $$g\mapsto\| b(g)\|$$ is called a Hilbert length function on $$G$$. In the paper under review the authors study some problems connected with the growth of Hilbert length functions. Discussing the existence of $$1$$-cocycles with linear growth, they 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, the authors discuss the converse, which turns out to hold for amenable groups with “controlled” Følner sequences; for general amenable groups they prove the weaker result that $$1$$-cocycles with sufficiently small growth are almost coboundaries. Besides, they show that there exist, on a-$$T$$-menable groups, proper cocycles with arbitrary small growth.

### MSC:

 22D10 Unitary representations of locally compact groups 43A07 Means on groups, semigroups, etc.; amenable groups 43A35 Positive definite functions on groups, semigroups, etc. 20F69 Asymptotic properties of groups
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