On the top-dimensional ℓ^2 -Betti numbers

Sur les nombres de Betti ℓ^2 en dimension maximale

Damien Gaboriau (1) , Camille Noûs (2)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 Laboratoire Cogitamus

The purpose of this note is to introduce a trick which relates the (non)-vanishing of the top-dimensional ℓ 2-Betti numbers of actions with that of sub-actions. We provide three different types of applications: we prove that the ℓ 2-Betti numbers of Aut(Fn) and Out(Fn) (and of their Torelli subgroups) do not vanish in degree equal to their virtual cohomological dimension, we prove that the subgroups of the 3-manifold groups have vanishing ℓ 2-Betti numbers in degree 3 and 2 and we prove for instance that F_2^d × Z has ergodic dimension d + 1.

Mots clés: L2-Betti numbers; ergodic dimension; Out(F n); Aut(F n); cohomological dimension; measured group theory; 3-dimensional manifolds

Domaines: Théorie des groupes [math.GR]

Damien Gaboriau, Camille Noûs. On the top-dimensional ℓ^2 -Betti numbers. Annales de la Faculté des Sciences de Toulouse. Mathématiques., In press. ⟨hal-02273797v2⟩