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.?
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$\begingroup$ which topology are you using for the convergence of automorphisms? $\endgroup$– Adrián González PérezCommented Dec 22, 2019 at 11:20
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$\begingroup$ Pointwise convergence in s.o.t. $\endgroup$– sibaniCommented Dec 23, 2019 at 9:57
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Such actions abound. For instance, one can embed $F_2$ in $\mathrm{SO}(3)$ (à la Banach–Tarski) and let the latter act on the hyperfinite $\mathrm{II}_1$-factor $R$ by realising the latter in terms of canonical anticommutation relations. 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).
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$\begingroup$ So, free group case can be extended it to any hyperbolic group. I understand the free group example. Can you please explain little about how to mimic the idea for any hyperbolic group? Thanks in advance. $\endgroup$– sibaniCommented Dec 23, 2019 at 11:04
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1$\begingroup$ Well, as written above, a residually finite hyperbolic group embeds densely into the profinite completion (which is a compact group and so embeds into $\mathrm{Aut}(R)$. But it is a big open problem whether every hyperbolic group is actually residually finite, so no example is known where one would need to come up with something else. $\endgroup$ Commented Dec 23, 2019 at 11:24