A representation $\pi: G \to B(H)$ of a group $G$ on a Hilbert space $H$ is called bounded iff $\sup_{g \in G} \| \pi(g) \|_{B(H)} = C < \infty$. A $1$-cocycle with respect to the representation $\pi$ is a map $\beta: G \to H$ such that $$ \beta(g \, h) = \beta(g) + \pi(g) \, \beta(h). $$ In the recent preprint
Nishikawa, Shintaro, $\mathrm{Sp}(n,1)$ admits a proper $1$-cocycle for a uniformly bounded representation, https://arxiv.org/abs/2003.03769.
It is claimed that $\mathrm{Sp}(1,n)$, although does not have the Haagerup approximation property, possess a metrically proper $1$-cocyle with respect to a bounded representation on a Hilbert space.
Question: Are there other known examples of groups without the Haagerup property (or with property $\mathrm{(T)}$) which admit a proper $1$-cocycler with respect to a bounded representation?
