[Sur la croissance homologique et les nombres de Betti $\ell ^2$ de $\operatorname{Out}(W_n)$)]
Let $n\ge 3$, and let $\operatorname{Out}(W_n)$ 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 $\operatorname{Out}(W_n)$. 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 $\operatorname{Out}(W_n)$ 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 $\lfloor \frac{n}{2}\rfloor -1$.
Soit $n\ge 3$, et soit $\operatorname{Out}(W_n)$ le groupe des automorphismes extérieurs d’un groupe de Coxeter libre $W_n$ de rang $n$. Nous étudions la croissance de la dimension des groupes d’homologie (à coefficients dans un corps $\mathbb{K}$ arbitraire) le long de suites de Farber de sous-groupes d’indice fini de $\operatorname{Out}(W_n)$. Nous montrons qu’en tout degré inférieur à $\lfloor \frac{n}{2}\rfloor -1$, ces nombres de Betti croissent sous-linéairement avec l’indice du sous-groupe. Lorsque $\mathbb{K}=\mathbb{Q}$, nous en déduisons grâce au théorème d’approximation de Lück que les nombres de Betti $\ell ^2$ de $\operatorname{Out}(W_n)$ s’annulent jusqu’en degré $\lfloor \frac{n}{2}\rfloor -1$. Par contraste, en dimension maximale $n-2$, un argument de Gaboriau et Noûs assure que le nombre de Betti $\ell ^2$ est non nul.
Nous démontrons aussi que la croissance de la torsion de l’homologie à coefficients entiers est sous-linéaire. Notre démonstration de ces résultats repose sur une méthode récemment introduite par Abért, Bergeron, Frączyk et Gaboriau. Un ingrédient-clé est de montrer qu’une version du complexe des bases partielles de $W_n$ a le type d’homotopie d’un bouquet de sphères de dimension $\lfloor \frac{n}{2}\rfloor -1$.
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Keywords: $\ell ^2$-Betti numbers, homology torsion growth, outer automorphisms of free Coxeter groups, outer space, complex of partial bases
Mots-clés : nombres de Betti $\ell ^2$, croissance de la torsion de l’homologie, automorphismes extérieurs de groupes de Coxeter libres, outre-espace, complexe des bases partielles
Gaboriau, Damien 1 ; Guerch, Yassine 2 ; Horbez, Camille 3
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author = {Gaboriau, Damien and Guerch, Yassine and Horbez, Camille},
title = {On the homology growth and the $\ell ^2${-Betti} numbers of $\operatorname{Out}(W_n)$},
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Gaboriau, Damien; Guerch, Yassine; Horbez, Camille. On the homology growth and the $\ell ^2$-Betti numbers of $\operatorname{Out}(W_n)$. Annales de l'Institut Fourier, Online first, 22 p.
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