Let $G$ be a hyperbolic group. Let $M$ be a vN algebra in standard form. Can there exist a faithful action of $G$ on $M$ such that 
\begin{align\*}
\sigma\_{g\_n} \rightarrow I
\end{align\*}
for some sequence $(g\_n)$ of hyperbolic elements.?



Action of hyperbolic group on von Neumann algebra
Such actions abound. For instance, one can embed $F\_2$ in $\mathrm{SO}(3)$ (à la [Banach–Tarski](https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox#A_sketch_of_the_proof)) and let the latter act on the hyperfinite $\mathrm{II}\_1$-factor $R$ by realising the latter in terms of [canonical anticommutation relations](https://mathoverflow.net/questions/178292/hyperfinite-type-ii-1-factor-as-the-clifford-algebra). The embedding $F\_2\to \mathrm{SO}(3)$ is dense and every non-trivial element is hyperbolic, so there will be a sequence as required.


If one reads your question as “does such an action exist for every fixed hyperbolic group $\Gamma$”, then I would still believe the answer to be yes, because at least in the residually finite case one can mimick the above construction, replacing $\mathrm{SO}(3)$ with the profinite completion of $\Gamma$ and embedding it into $\mathrm{Aut}(R)$ (in view of the CAR description, the latter is known to contain every separable locally compact group).


