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

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

2008-03-13

arxiv.org/abs/0803.2062v3

Mathematics

Group Theory

Typos corrected, reference and thanks added. Final version, to appear in Commetarii. Math. Helv

Scientific paper

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.

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