Isometric group actions on Hilbert spaces: growth of cocycles (Q2458410)

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 \textit{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.