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Let $n\ge 3$, and let $\mathrm{Out}(W_n)$ be the outer automorphism group ofa free Coxeter group $W_n$ of rank $n$. We study the growth of the dimension ofthe homology groups (with coefficients in any field $\mathbb{K}$) along Farbersequences of finite-index subgroups of $\mathrm{Out}(W_n)$. We show that, inall degrees up to $\lfloor\frac{n}{2}\rfloor-1$, these Betti numbers growsublinearly in the index of the subgroup. When $\mathbb{K}=\mathbb{Q}$, throughL\"uck's approximation theorem, this implies that all $\ell^2$-Betti numbers of$\mathrm{Out}(W_n)$ vanish up to degree $\lfloor\frac{n}{2}\rfloor-1$. Incontrast, in top dimension equal to $n-2$, an argument of Gaboriau and No\^usimplies that the $\ell^2$-Betti number does not vanish. We also prove that thetorsion growth of the integral homology is sublinear. Our proof of theseresults relies on a recent method introduced by Ab\'ert, Bergeron, Fr\k{a}czykand Gaboriau. A key ingredient is to show that a version of the complex ofpartial bases of $W_n$ has the homotopy type of a bouquet of spheres ofdimension $\lfloor\frac{n}{2}\rfloor-1$.

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