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

Martin R. Bridson and Karen Vogtmann

Version of 12 March 2008. Minor edits 15 October 2008. To appear in Commentarii Math Helv.

For $n\ge 3$, let $SAut(F_n)$ denote the unique subgroup of index two in the automorphism group of a free group of rank $n$.

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.