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

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.Comment: Typos corrected, reference and thanks added. Final version, to appear in Commetarii. Math. Hel

Abelian covers of graphs and maps between outer automorphism groups of free groups

We explore the existence of homomorphisms between outer automorphism groups of free groups Out(F_n) \to Out(F_m). We prove that if n > 8 is even and n \neq m \leq 2n, or n is odd and n \neq m \leq 2n - 2, then all such homomorphisms have finite image; in fact they factor through det: Out(F_n) \to Z/2. In contrast, if m = r^n(n - 1) + 1 with r coprime to (n - 1), then there exists an embedding Out(F_n) \to Out(F_m). In order to prove this last statement, we determine when the action of Out(F_n) by homotopy equivalences on a graph of genus n can be lifted to an action on a normal covering with abelian Galois group.Comment: Final version, to appear in Mathematische Annalen. Minor errors and typos corrected, including range of n in Theorem

The Dehn functions of Out(F_n) and Aut(F_n)

For n > 2, the Dehn functions of Aut(F_n) and Out(F_n) are exponential. Hatcher and Vogtmann proved that they are at most exponential, and the complementary lower bound in the case n=3 was established by Bridson and Vogtmann. Handel and Mosher completed the proof by reducing the lower bound for n>4 to the case n=3. In this note we give a shorter, more direct proof of this last reduction.Comment: Final version, to appear in Annales de l'Institut Fourie