On the top-dimensional ℓ2 -Betti numbers

Damien Gaboriau, Camille Noûs

Preprint, to appear in Annales de la Faculté des Sciences de Toulouse.

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Hal: https://hal.archives-ouvertes.fr/hal-02273797
arXiv : https://arxiv.org/abs/1909.01633

Abstract

The purpose of this note is to introduce a trick which relates the (non)-vanishing of the top-dimensional ℓ2-Betti numbers of actions with that of sub-actions. We provide three different types of applications: we prove that the ℓ2-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 ℓ2-Betti numbers in degree 3 and 2 and we prove for instance that F2d × Z has ergodic dimension d + 1.

With Camille Noûs,  from the Laboratory "Cogitamus"

What is the meaning of Camille Noûs ? (Curriculum Vitæ of Camille Noûs)

"Camille Noûs" is a scientific consortium created to affirm the collaborative and open nature of knowledge creation and dissemination, under the control of the academic community. This scientific collective, like Bourbaki, Henri Paul de Saint Gervais or Arthur Besse in mathematics, or Isadore Nabi in biology, takes on the identity of a scientific personality who embodies the collective contribution of the academic community. More precisely, Camille Noûs is a collective individual who symbolizes our deep attachment to the values of ethics and probation that are carried by the contradictory debate, she is insensitive to the indicators elaborated by the institutional management of research, she knows what our results owe to collective construction. This is the meaning of the "Noûs", bearing a collegial We but referring above all to the concept of "reason" (or "rational" or "intellect") inherited from Greek philosophy.