The paper [The Tits alternative for $\operatorname{Out}(F\_n)$ I](https://arxiv.org/pdf/math/9712217.pdf) by Bestvina, Feighn and Handel and the paper [Automorphisms of free groups and Outer space](http://www.math.cornell.edu/~vogtmann/papers/Autosurvey/autosurvey.pdf) by Vogtmann both state that $\operatorname{Aut}(F\_n)$ and $\operatorname{Out}(F\_n)$ are not linear groups for $n \geq 3$ respectively $n \geq 4$. They both cite [The automorphism group of a free group is not linear](https://www.sciencedirect.com/science/article/pii/002186939290029L) by Formanek and Procesi. This text proves the claim about the automorphism group of a free group, but does not mention the claim about the outer automorphism group. 


I was wondering if some properties about linear groups imply this result, but I couldn't figure out what property would. Any help would be appreciated.


**Once again, not really sure if this is the right place to post this question. If not, I'll take it down (or if someone knows how to relocate it to mathstack, that would be great too)**



$\operatorname{Out}(F_n)$ is not linear for $n > 3$
There is an embedding $\text{Aut}(F\_{n-1}) \hookrightarrow \text{Out}(F\_n)$ for any $n \ge 2$, as follows. 


First one embeds $\text{Aut}(F\_{n-1}) \hookrightarrow \text{Aut}(F\_n)$ by extending any automorphism of $F\_{n-1}$ to an automorphism of 
$$F\_n \approx F\_{n-1} \* \mathbb{Z}
$$
using the identity automorphism of $\mathbb{Z}$. 


Next one notices that none of the resulting automorphisms of $F\_n$ are inner, hence we get an embedding $\text{Aut}(F\_{n-1}) \hookrightarrow \text{Out}(F\_n)$. 


It follows that if $\text{Out}(F\_n)$ is linear then $\text{Aut}(F\_{n-1})$ is linear.


