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.



Genericity of irreducible automorphisms of free groups
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.


