A substitute for Kazhdan's property (T) for universal non-lattices
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914925091225600 |
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| author | Ozawa, Narutaka |
| author_facet | Ozawa, Narutaka |
| contents | The well-known theorem of Shalom--Vaserstein and Ershov--Jaikin-Zapirain states that the group $\mathrm{EL}_n(\mathcal{R})$, generated by elementary matrices over a finitely generated commutative ring $\mathcal{R}$, has Kazhdan's property (T) as soon as $n\geq3$. This is no longer true if the ring $\mathcal{R}$ is replaced by a commutative rng (a ring but without the identity) due to nilpotent quotients $\mathrm{EL}_n(\mathcal{R}/\mathcal{R}^k)$. In this paper, we prove that even in such a case the group $\mathrm{EL}_n(\mathcal{R})$ satisfies a certain property that can substitute property (T), provided that $n$ is large enough. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_05272 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A substitute for Kazhdan's property (T) for universal non-lattices Ozawa, Narutaka Functional Analysis Group Theory 22D10, 46L89, 22D15 The well-known theorem of Shalom--Vaserstein and Ershov--Jaikin-Zapirain states that the group $\mathrm{EL}_n(\mathcal{R})$, generated by elementary matrices over a finitely generated commutative ring $\mathcal{R}$, has Kazhdan's property (T) as soon as $n\geq3$. This is no longer true if the ring $\mathcal{R}$ is replaced by a commutative rng (a ring but without the identity) due to nilpotent quotients $\mathrm{EL}_n(\mathcal{R}/\mathcal{R}^k)$. In this paper, we prove that even in such a case the group $\mathrm{EL}_n(\mathcal{R})$ satisfies a certain property that can substitute property (T), provided that $n$ is large enough. |
| title | A substitute for Kazhdan's property (T) for universal non-lattices |
| topic | Functional Analysis Group Theory 22D10, 46L89, 22D15 |
| url | https://arxiv.org/abs/2207.05272 |