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

TL;DR

The paper proves that the automorphism groups of free groups, specifically SAut(F_n), cannot act non-trivially on low-dimensional acyclic manifolds or spheres, extending to SL(n,Z) and considering different coefficients.

Contribution

It establishes new non-existence results for group actions of SAut(F_n) on certain low-dimensional topological spaces, including spheres and acyclic manifolds.

Findings

SAut(F_n) has no non-trivial actions on acyclic manifolds of dimension less than n.

SAut(F_n) cannot act non-trivially on spheres of dimension less than n-1.

Results extend to SL(n,Z) and Z_3 coefficients when n is even.

Abstract

For n at least 3, let SAut(F_n) denote the unique subgroup of index two in the automorphism group of a free group. The standard linear action of SL(n,Z) on R^n induces non-trivial actions of SAut(F_n) on R^n and on S^{n-1}. We prove that SAut(F_n) admits no non-trivial actions by homeomorphisms on acyclic manifolds or spheres of smaller dimension. Indeed, SAut(F_n) 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.