# On the homology growth and the ℓ2 -Betti numbers of Out(Wn)

## Damien Gaboriau, Yassine Guerch and Camille Horbez

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Hal: https://hal.archives-ouvertes.fr/hal-03775219v1
arXiv: https://arxiv.org/abs/2209.02760

## Abstract

Let n3, and let Out(Wn) be the outer automorphism group of a free Coxeter group Wn 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(Wn). We show that, in all degrees up to n21, these Betti numbers grow sublinearly in the index of the subgroup. When 𝕂=, through Lück's approximation theorem, this implies that all 2-Betti numbers of Out(Wn) vanish up to degree n21. In contrast, in top dimension equal to n2, an argument of Gaboriau and Noûs implies that the 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, Frączyk and Gaboriau. A key ingredient is to show that a version of the complex of partial bases of Wn has the homotopy type of a bouquet of spheres of dimension n21.