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 , let denote the unique subgroup of index two in the automorphism group of a free group. The standard linear action of on induces non-trivial actions of on and on . We prove that admits no non-trivial actions by homeomorphisms on acyclic manifolds or spheres of smaller dimension. Indeed, cannot act non-trivially on any generalized -homology sphere of dimension less than , nor on any -acyclic -homology manifold of dimension less than . It follows that cannot act non-trivially on such spaces either. When is even, we obtain similar results with 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