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

Abstract

For $n\ge 3$, let $\mathrm{SAut}(F_n)$ denote the unique subgroup of index two in the automorphism group of a free group. The standard linear action of $\mathrm{SL}(n,Z)$ on $R^n$ induces non-trivial actions of $\mathrm{SAut}(F_n)$ on $R^n$ and on ${\mathbb{S}}^{n-1}$. We prove that $\mathrm{SAut}(F_n)$ admits no non-trivial actions by homeomorphisms on acyclic manifolds or spheres of smaller dimension. Indeed, $\mathrm{SAut}(F_n)$ 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 $\mathrm{SL}(n,Z)$ cannot act non-trivially on such spaces either. When $n$ is even, we obtain similar results with $Z_3$ coefficients.