Non-vanishing unitary cohomology of low-rank integral special linear groups
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914362076168192 |
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| author | Brück, Benjamin Hughes, Sam Kielak, Dawid Mizerka, Piotr |
| author_facet | Brück, Benjamin Hughes, Sam Kielak, Dawid Mizerka, Piotr |
| contents | We construct explicit finite-dimensional orthogonal representations $π_N$ of $\operatorname{SL}_{N}(\mathbb{Z})$ for $N \in \{3,4\}$ all of whose invariant vectors are trivial, and such that $H^{N - 1}(\operatorname{SL}_{N}(\mathbb{Z}),π_N)$ is non-trivial. This implies that for $N$ as above, the group $\operatorname{SL}_{N}(\mathbb{Z})$ does not have property $(T_{N-1})$ of Bader-Sauer and therefore is not $(N-1)$-Kazhdan in the sense of De Chiffre-Glebsky-Lubotzky-Thom, both being higher versions of Kazhdan's property $T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22310 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-vanishing unitary cohomology of low-rank integral special linear groups Brück, Benjamin Hughes, Sam Kielak, Dawid Mizerka, Piotr Group Theory Algebraic Topology 11F75, 20J06, 55-08 We construct explicit finite-dimensional orthogonal representations $π_N$ of $\operatorname{SL}_{N}(\mathbb{Z})$ for $N \in \{3,4\}$ all of whose invariant vectors are trivial, and such that $H^{N - 1}(\operatorname{SL}_{N}(\mathbb{Z}),π_N)$ is non-trivial. This implies that for $N$ as above, the group $\operatorname{SL}_{N}(\mathbb{Z})$ does not have property $(T_{N-1})$ of Bader-Sauer and therefore is not $(N-1)$-Kazhdan in the sense of De Chiffre-Glebsky-Lubotzky-Thom, both being higher versions of Kazhdan's property $T$. |
| title | Non-vanishing unitary cohomology of low-rank integral special linear groups |
| topic | Group Theory Algebraic Topology 11F75, 20J06, 55-08 |
| url | https://arxiv.org/abs/2410.22310 |