On the homology growth and the L^2-Betti numbers of Out(W_n)

Damien Gaboriau (1, 2, 3) , Yassine Guerch (4, 5) , Camille Horbez (1, 4, 5)

1 CNRS - Centre National de la Recherche Scientifique
2 ENS de Lyon - École normale supérieure de Lyon
3 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
4 Université Paris-Saclay
5 LMO - Laboratoire de Mathématiques d'Orsay

Let n ≥ 3, and let Out(W_n) be the outer automorphism group of a free Coxeter group W_n of rank n. We study the growth of the dimension of the homology groups (with coefficients in any field K) along Farber sequences of finite-index subgroups of Out(W_n). We show that, in all degrees up to n/2 - 1, these Betti numbers grow sublinearly in the index of the subgroup. When K=Q, through Lueck's approximation theorem, this implies that all L^2-Betti numbers of Out(W_n) vanish up to degree floor(n/2 -1). In contrast, in top dimension equal to n-2, an argument of Gaboriau and Noûs implies that the L^2-Betti number does not vanish. We also prove that the torsion growth of the integral homology is sublinear. Our proof of these results relies on a recent method introduced by Abért, Bergeron, Fraczyk and Gaboriau. A key ingredient is to show that a version of the complex of partial bases of W_n has the homotopy type of a bouquet of spheres of dimension floor(n/2 -1).

Domaines: Théorie des groupes [math.GR]; Topologie algébrique [math.AT]; Topologie géométrique [math.GT]

Damien Gaboriau, Yassine Guerch, Camille Horbez. On the homology growth and the L^2-Betti numbers of Out(W_n). Annales de l'Institut Fourier, In press. ⟨hal-03775219⟩