#### Algebraic and Geometric Topology 4 (2004),
paper no. 54, pages 1253-1272.

## Homology stability for outer automorphism groups of free groups

### Allen Hatcher, Karen Vogtmann

**Abstract**.
We prove that the quotient map from Aut(F_n) to Out(F_n) 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(F_n), showing that its i-th homology is independent of n
for n at least 2i+2.
**Keywords**.
Automorphisms of free groups, homology stability

**AMS subject classification**.
Primary: 20F65.
Secondary: 20F28, 57M07.

**DOI:** 10.2140/agt.2004.4.1253

**E-print:** `arXiv:math.GT/0406377`

Submitted: 28 June 2004.
Accepted: 7 December 2004.
Published: 24 December 2004.

Notes on file formats
Allen Hatcher, Karen Vogtmann

Department of Mathematics, Cornell University

Ithaca, NY 14853-4201, USA

Email: hatcher@math.cornell.edu, vogtmann@math.cornell.edu

AGT home page

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