On the top-dimensional $\ell ^2$-Betti numbers

Damien Gaboriau; Camille Noûs

doi:10.5802/afst.1695

On the top-dimensional

ℓ^{2}

-Betti numbers

Damien Gaboriau ¹; Camille Noûs ²

¹ CNRS & U.M.P.A., Ecole Normale Supérieure de Lyon, UMR 5669, 46 allée d’Italie, 69364 Lyon cedex 07, France
² Laboratoire Cogitamus

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 5, pp. 1121-1137.

Abstract
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 $Aut (F_{n})$ and $Out (F_{n})$ (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 figure out the ergodic dimension of certain direct products of the form $H \times A$ where $A$ is infinite amenable.

Le but de cette note est d’introduire une astuce qui relie l’annulation (ou la non-annulation) du nombre de Betti $ℓ^{2}$ en dimension maximale des actions d’un groupe avec l’annulation pour ses sous-actions. On fournit trois différents types d’applications : on montre que les nombres de Betti $ℓ^{2}$ de $Aut (F_{n})$ et $Out (F_{n})$ (et de leurs sous-groupes de Torelli) ne s’annulent pas en degré égal à leur dimension cohomologique virtuelle ; on prouve qu’un sous-groupe quelconque du groupe fondamental d’une variété compacte de dimension $3$ a ses nombres de Betti $ℓ^{2}$ nuls en degré $3$ et $2$ et enfin, on parvient à déterminer la dimension ergodique de certains produits directs de la forme $H \times A$ où $A$ est moyennable infini.

Received: 2019-10-13
Accepted: 2020-05-11
Published online: 2022-03-11

DOI: 10.5802/afst.1695

Classification: 37A20, 19K56, 20F28, 20E15, 57MXX
Keywords: $\ell ^2$-Betti numbers, measured group theory, cohomological dimension, ergodic dimension, $\mathrm{Out}(\mathbf{F}_n), \mathrm{Aut}(\mathbf{F}_n)$, $3$-dimensional manifolds.

Author's affiliations:

Damien Gaboriau ¹; Camille Noûs ²

¹ CNRS & U.M.P.A., Ecole Normale Supérieure de Lyon, UMR 5669, 46 allée d’Italie, 69364 Lyon cedex 07, France
² Laboratoire Cogitamus

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Damien Gaboriau; Camille Noûs. On the top-dimensional $\ell ^2$-Betti numbers. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 5, pp. 1121-1137. doi : 10.5802/afst.1695. https://afst.centre-mersenne.org/articles/10.5802/afst.1695/

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