On the top-dimensional $\ell^2$-Betti numbers
| Autor: | Gaboriau, Damien, Noûs, Camille |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques, Serie 6, Volume 30 (2021) no. 5, pp. 1121-1137 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.5802/afst.1695 |
| Popis: | The purpose of this note is to introduce a trick which relates the (non)-vanishing of the top-dimensional $\ell^2$-Betti numbers of actions with that of sub-actions. We provide three different types of applications: we prove that the $\ell^2$-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 $\ell^2$-Betti numbers in degree 3 and 2 and we prove for instance that $F_2^d \times Z$ has ergodic dimension $d + 1$. Comment: ''Camille No\^us'' 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, H. P. de Saint Gervais or A. Besse in mathematics takes on the identity of a scientific personality who embodies the collective contribution of the academic community |
| Databáze: | arXiv |
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