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

Abstract

For n ≥ 3, let SAut(Fn) denote the unique subgroup of index two in the automorphism group of a free group. The standard linear action of SL(n,ℤ) on ℝn induces non-trivial actions of SAut(Fn) on ℝn and on Sn−1. We prove that SAut(Fn) admits no non-trivial actions by homeomorphisms on acyclic manifolds or spheres of smaller dimension. Indeed, SAut(Fn) cannot act non-trivially on any generalized ℤ2-homology sphere of dimension less than n − 1, nor on any ℤ2-acyclic ℤ2-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 ℤ3 coefficients.