On the homology growth and
-Betti numbers of Out(Wn)
Damien Gaboriau, Yassine Guerch and Camille Horbez
and let Out(Wn) be the
outer automorphism group of a free Coxeter group Wn
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(Wn). We
show that, in all degrees up to ⌊n2⌋−1,
these Betti numbers grow sublinearly in the index of the subgroup.
through Lück's approximation theorem, this implies that all ℓ2-Betti
numbers of Out(Wn) vanish
up to degree ⌊n2⌋−1.
In contrast, in top dimension equal to n−2,
an argument of Gaboriau and Noûs implies that the ℓ2-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 Wn
has the homotopy type of a bouquet of spheres of dimension ⌊n2⌋−1.