#### Volume 4, issue 2 (2004)

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Homology stability for outer automorphism groups of free groups

### Allen Hatcher and Karen Vogtmann

Algebraic & Geometric Topology 4 (2004) 1253–1272
 arXiv: math.GT/0406377
##### 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 $2i+2$.

##### Keywords
automorphisms of free groups, homology stability
##### Mathematical Subject Classification 2000
Primary: 20F65
Secondary: 20F28, 57M07