# Actions of automorphism groups of free groups on homology spheres and acyclic manifolds

### Martin R. Bridson

University of Oxford, UK### Karen Vogtmann

University of Warwick, Coventry, United Kingdom

## Abstract

For $n≥3$, let $SAut(F_{n})$ denote the unique subgroup of index two in the automorphism group of a free group. The standard linear action of $SL(n,Z)$ on $R_{n}$ induces non-trivial actions of $SAut(F_{n})$ on $R_{n}$ and on $S_{n−1}$. We prove that $SAut(F_{n})$ admits no non-trivial actions by homeomorphisms on acyclic manifolds or spheres of smaller dimension. Indeed, $SAut(F_{n})$ cannot act non-trivially on any generalized $Z_{2}$-homology sphere of dimension less than $n−1$, nor on any $Z_{2}$-acyclic $Z_{2}$-homology manifold of dimension less than $n$. It follows that $SL(n,Z)$ cannot act non-trivially on such spaces either. When $n$ is even, we obtain similar results with $Z_{3}$ coefficients.

## Cite this article

Martin R. Bridson, Karen Vogtmann, Actions of automorphism groups of free groups on homology spheres and acyclic manifolds. Comment. Math. Helv. 86 (2011), no. 1, pp. 73–90

DOI 10.4171/CMH/218