Stabilité homologique pour les groupes d'automorphismes des produits libres

Abstract

We show in this article that, for any group $G$ indecomposable for the free product * and non-isomorphic to $\mathbf{Z}$, the canonical inclusion $\mathrm{Aut}(G^{\ast n})\to \mathrm{Aut}(G^{\ast n+1})$ induces an isomorphism between the homology groups $H_i$ for $n\ge 2i+2$, 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.

arXiv comment17 pages

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