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

TL;DR

This paper investigates the homology and $ ext{l}^2$-Betti numbers of the outer automorphism group of free Coxeter groups, showing vanishing results in lower degrees and non-vanishing in top degree, using recent topological methods.

Contribution

It establishes new vanishing and non-vanishing results for $ ext{l}^2$-Betti numbers of $ ext{Out}(W_n)$ up to a specific degree, applying a novel homotopy analysis of a related complex.

Findings

Betti numbers grow sublinearly in all degrees up to $rac{n}{2}-1$

All $ ext{l}^2$-Betti numbers vanish in degrees up to $rac{n}{2}-1$

Non-vanishing of $ ext{l}^2$-Betti number in top degree $n-2$

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 L\"uck's approximation theorem, this implies that all $\ell^2$-Betti numbers of $\mathrm{Out}(W_n)$ vanish up to degree $\lfloor\frac{n}{2}\rfloor-1$. In contrast, in top dimension equal to $n-2$, an argument of Gaboriau and No\^us 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 Ab\'ert, Bergeron, Fr\k{a}czyk 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 $\lfloor\frac{n}{2}\rfloor-1$.