Stabilité homologique pour les groupes d'automorphismes des produits libres

Details Stabilité homologique pour les groupes d'automorphismes des produits libres Stabilité homologique pour les groupes d'automorphismes des produits libres

2011-09-13

arxiv.org/abs/1109.2686v1

Mathematics

Algebraic Topology

17 pages

Scientific paper

We show in this article that, for any group $G$ indecomposable for the free product * and non-isomorphic to $\mathbf{Z}$, the canonical inclusion ${\rm Aut}(G^{*n})\to {\rm Aut}(G^{* n+1})$ induces an isomorphism between the homology groups $H_i$ for $n\geq 2i+2$, as was conjectured by Hatcher and Wahl. In fact we show a little more --- in particular, the result is true for any group $G$ if we replace the automorphism group of the free product by the subgroup of symmetric automorphisms. For this purpose we use constructions and acyclicity results due to McCullough-Miller and Chen-Glover-Jensen and functoriality properties which allow us to apply classical methods in functor homology.

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