On the homology growth and the ${\ell }^2$-Betti numbers of $\mathrm{Out}(W_n)$

Abstract

Let $n\ge 3$, and let $\mathrm{Out}(W_n)$ 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 $\mathbb{K}$) along Farber sequences of finite-index subgroups of $\mathrm{Out}(W_n)$. We show that, in all degrees up to $\lfloor \frac{n}{2}\rfloor -1$, these Betti numbers grow sublinearly in the index of the subgroup. When $\mathbb{K}=\mathbb{Q}$, through Lück's approximation theorem, this implies that all ${\ell }^2$-Betti numbers of $\mathrm{Out}(W_n)$ vanish up to degree $\lfloor \frac{n}{2}\rfloor -1$. In contrast, in top dimension equal to $n-2$, an argument of Gaboriau and Noûs implies that the ${\ell }^2$-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 $W_n$ has the homotopy type of a bouquet of spheres of dimension $\lfloor \frac{n}{2}\rfloor -1$.