I am not an expert at all in the subject of Lie groups, lattices, arithmetic groups and rigidity. But, lately I am interested in Margulis superrigidity theorem, which in most versions can be stated as follows:

Theorem. *Let $G$ and $G'$ be semisimple connected real center-free Lie groups without compact factors with $\mathrm{rk}(G)\geq 2$, $\Gamma < G$ be an irreducible lattice, and $\pi: \Gamma \to G'$ a homomorphism with $\pi(\Gamma)$ being Zariski dense in $G'$. Then $\pi$ extends to a rational epimorphism $\pi':G\to G'$.*

Here "$H$ is without compact factors" means, for $H$ center-free, that if we write $H=\prod_{i=1}^kS_i$ with $S_i$ simple, then each $S_i$ is non-compact, or equivalently has positive (real) rank; the (real) rank $\mathrm{rk}(H)$ of $H$ is the sum of the ranks of all $S_i$. Irreducibility of a lattice $\Gamma$ means that its projection in $H/S_i$ has a dense image for all $i$.

Questions:

Do you have examples of Lie groups and lattices for which Margulis theorem applies, and also groups for which the theorem does not holds. In particular, do this theorem apply for $G=G'=\mathrm{PSL}(n,\mathbb{R})$ and $\Gamma=\mathrm{PSL}(n,\mathbb{Z})$. Does this implies that $\mathrm{Out}(\mathrm{PSL}(n,\mathbb{Z}))$ is finite?

Thank you all.

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