Commentarii Mathematici Helvetici


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Volume 86, Issue 1, 2011, pp. 73–90
DOI: 10.4171/CMH/218

Published online: 2010-12-05

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

Martin R. Bridson[1] and Karen Vogtmann[2]

(1) University of Oxford, UK
(2) University of Warwick, Coventry, United Kingdom

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 Sn−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.

Keywords: Automorphism groups of free groups, rigidity, group actions, Smith theory, generalized manifolds

Bridson Martin, Vogtmann Karen: Actions of automorphism groups of free groups on homology spheres and acyclic manifolds. Comment. Math. Helv. 86 (2011), 73-90. doi: 10.4171/CMH/218