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I have seen in the literature that Irreducible outer automorphisms of a free group $F_n$ are "generic".

I would like to ask that if for example : it is true that for any generating set $X$ of $Out(F_n)$ if we take the $k$- closed ball in the Cayley graph (with respect to $X$) $B(k)$ and we denote by $R$ the set of reducible automorphisms can we get something like :

For $n$ goes to infinity$, \frac{| R \cap B(k) |}{| B(k)|}$ goes to 0 ? (by | | I mean the number of elements of the group that each subset contains)

I have seen that the genericity is given in terms of random walks and random elements, but I don't know if these notions imply that or if there is something like this in the literature.

Thanks a lot for your time.

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No, this is not known. The results for random elements (which say that a random autormorphism is irreducible with irreducible powers (iwip)) sadly do not imply this.

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