Homology stability for outer automorphism groups of free groups

Abstract

We prove that the quotient map from $Aut\left(F_n\right)$ to $Out\left(F_n\right)$ induces an isomorphism on homology in dimension $i$ for $n$ at least $2i+4$. This corrects an earlier proof by the first author and significantly improves the stability range. In the course of the proof, we also prove homology stability for a sequence of groups which are natural analogs of mapping class groups of surfaces with punctures. In particular, this leads to a slight improvement on the known stability range for $Aut\left(F_n\right)$, showing that its $i^{th}$ homology is independent of $n$ for $n$ at least $2i+2$.