Stabilit\'e homologique pour les groupes d'automorphismes des produits libres

TL;DR

This paper proves that for certain groups, the homology of automorphism groups stabilizes as the number of free factors increases, confirming a conjecture by Hatcher and Wahl and extending to symmetric automorphisms.

Contribution

It establishes homological stability for automorphism groups of free products of groups, including symmetric automorphisms, using functor homology techniques.

Findings

Homology groups stabilize for groups with multiple free factors.

Stability holds for symmetric automorphisms beyond original conjecture.

Uses constructions and acyclicity results from McCullough-Miller and Chen-Glover-Jensen.

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 ${\rm Aut}(G^{*n})\to {\rm Aut}(G^{* n+1})$ 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.