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Ergodic theorems for affine actions of amenable groups on Hilbert space

We prove a new weak mean ergodic theorem (Theorem A) for 1-cocycles associated to weakly mixing representations of amenable groups. Let $G$ be a finitely generated, discrete, amenable group $G$ which admits a controlled Folner sequence. We use Theorem A to deduce that any affine action $G\ca^T \Cal H$ on Hilbert space with weakly mixing linear part admits a sequence of almost fixed points (Theorem B). Specializing to the case that $G$ is a finitely generated group of polynomial growth, we show that convex combinations of averages of the associated 1-cocycle over $n$-balls provide a sequence of almost fixed points for the action $G\ca^T \Cal H$ (Corollary C). This affirms a weak form of a conjecture of Shalom independently of Gromov's theorem on the virtual nilpotency of groups of polynomial growth. As a consequence, we are able to give a new, elementary, ergodic-theoretical proof of Gromov's theorem.