On vanishing criteria of $L^2$-Betti numbers of groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909069212647424 |
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| author | Sánchez-Peralta, Pablo |
| author_facet | Sánchez-Peralta, Pablo |
| contents | Let $G$ be a countable group and $k$ a positive integer, we show that the $L^2$-Betti numbers of the group $G$ vanish up to degree $k$ provided that there is some infinite index subgroup $H$ with finite $k$th $L^2$-Betti number containing a normal subgroup of $G$ whose $L^2$-Betti numbers are all zero below degree $k$. This generalizes previous criteria of both Sauer and Thom, and Peterson and Thom. In addition, we exhibit a purely algebraic proof of a well-known theorem of Gaboriau concerning the first $L^2$-Betti number which was requested by Bourdon, Martin and Valette. Finally, we provide evidence of a positive answer for a question posted by Hillman that wonders whether the above statement holds for $k = 1$ and $H$ containing a subnormal subgroup instead. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_07031 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On vanishing criteria of $L^2$-Betti numbers of groups Sánchez-Peralta, Pablo Group Theory Dynamical Systems 20J05, 37A20, 46L99 Let $G$ be a countable group and $k$ a positive integer, we show that the $L^2$-Betti numbers of the group $G$ vanish up to degree $k$ provided that there is some infinite index subgroup $H$ with finite $k$th $L^2$-Betti number containing a normal subgroup of $G$ whose $L^2$-Betti numbers are all zero below degree $k$. This generalizes previous criteria of both Sauer and Thom, and Peterson and Thom. In addition, we exhibit a purely algebraic proof of a well-known theorem of Gaboriau concerning the first $L^2$-Betti number which was requested by Bourdon, Martin and Valette. Finally, we provide evidence of a positive answer for a question posted by Hillman that wonders whether the above statement holds for $k = 1$ and $H$ containing a subnormal subgroup instead. |
| title | On vanishing criteria of $L^2$-Betti numbers of groups |
| topic | Group Theory Dynamical Systems 20J05, 37A20, 46L99 |
| url | https://arxiv.org/abs/2307.07031 |