On the homology growth and the ℓ 2 -Betti numbers of Out ( W n )

Let n ≥ 3 , and let 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 𝕂 ) along Farber sequences of finite-index subgroups of Out ( W n ) . We show that, in all degrees up to ⌊ n 2 ⌋ - 1 , these Betti numbers grow sublinearly in the index of the subgroup. When 𝕂 = ℚ , through Lück’s approximation theorem, this implies that all ℓ 2 -Betti numbers of Out ( W n ) vanish up to degree ⌊ n 2 ⌋ - 1 . In contrast, in top dimension equal to n - 2 , an argument of Gaboriau and Noûs implies that the ℓ 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 ⌊ n 2 ⌋ - 1 .