L^2-Betti numbers arising from the lamplighter group
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| Format: | Preprint |
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2020
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| author | Ara, Pere Claramunt, Joan |
| author_facet | Ara, Pere Claramunt, Joan |
| contents | We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute $\ell^2$-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational $\ell^2$-Betti numbers arising from the lamplighter group algebra $K[\mathbb{Z}_2 \wr \mathbb{Z}]$, being $K$ a subfield of the complex numbers closed under complex conjugation. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational $\ell^2$-Betti numbers from the algebras $\mathbb{Q}[\mathbb{Z}_n \wr \mathbb{Z}]$, where $n \geq 2$ is a natural number. We also apply the techniques developed to the (generalized) odometer algebra $\mathcal{O}(\overline{n})$, where $\overline{n}$ is a supernatural number. We compute its $*$-regular closure, and this allows us to fully characterize the set of $\ell^2$-Betti numbers arising from $\mathcal{O}(\overline{n})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2005_12814 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | L^2-Betti numbers arising from the lamplighter group Ara, Pere Claramunt, Joan Group Theory Combinatorics Rings and Algebras Primary 20C07, Secondary 20F65, 16E50 We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute $\ell^2$-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational $\ell^2$-Betti numbers arising from the lamplighter group algebra $K[\mathbb{Z}_2 \wr \mathbb{Z}]$, being $K$ a subfield of the complex numbers closed under complex conjugation. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational $\ell^2$-Betti numbers from the algebras $\mathbb{Q}[\mathbb{Z}_n \wr \mathbb{Z}]$, where $n \geq 2$ is a natural number. We also apply the techniques developed to the (generalized) odometer algebra $\mathcal{O}(\overline{n})$, where $\overline{n}$ is a supernatural number. We compute its $*$-regular closure, and this allows us to fully characterize the set of $\ell^2$-Betti numbers arising from $\mathcal{O}(\overline{n})$. |
| title | L^2-Betti numbers arising from the lamplighter group |
| topic | Group Theory Combinatorics Rings and Algebras Primary 20C07, Secondary 20F65, 16E50 |
| url | https://arxiv.org/abs/2005.12814 |