| Actions of automorphism groups of free groups on homology spheres and acyclic manifolds |
| Martin R. Bridson and Karen Vogtmann |
| submitted |
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 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. |