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On the top-dimensional ℓ^2 -Betti numbers

Abstract : 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.
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Contributor : Damien Gaboriau <>
Submitted on : Thursday, June 11, 2020 - 3:44:57 PM
Last modification on : Friday, June 26, 2020 - 6:13:51 PM

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  • HAL Id : hal-02273797, version 2
  • ARXIV : 1909.01633

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

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