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

### Martin R. Bridson

University of Oxford, UK### Karen Vogtmann

University of Warwick, Coventry, United Kingdom

## 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 **S**n−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.

## Cite this article

Martin R. Bridson, Karen Vogtmann, Actions of automorphism groups of free groups on homology spheres and acyclic manifolds. Comment. Math. Helv. 86 (2011), no. 1, pp. 73–90

DOI 10.4171/CMH/218