Actions of automorphism groups of free groups on homology spheres and acyclic manifolds
Abstract
For n at least 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.
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