Inducing spectral gaps for the cohomological Laplacians of $\operatorname{SL}_n(\mathbb{Z})$ and $\operatorname{SAut}(F_n)$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915254179463168 |
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| author | Mizerka, Piotr |
| author_facet | Mizerka, Piotr |
| contents | The technique of inducing spectral gaps for cohomological Laplacians in degree zero was used by Kaluba, Kielak and Nowak to prove property (T) for $\operatorname{SAut}(F_n)$ and $\operatorname{SL}_n(\mathbb{Z})$. In this paper, we adapt this technique to Laplacians in degree one. This allows to provide a lower bound for the cohomological Laplacian in degree one for $\operatorname{SL}_n(\mathbb{Z})$ for every unitary representation. In particular, one gets in that way an alternative proof of property (T) for $\operatorname{SL}_n(\mathbb{Z})$ whenever $n\geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10287 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inducing spectral gaps for the cohomological Laplacians of $\operatorname{SL}_n(\mathbb{Z})$ and $\operatorname{SAut}(F_n)$ Mizerka, Piotr Group Theory 20G10 The technique of inducing spectral gaps for cohomological Laplacians in degree zero was used by Kaluba, Kielak and Nowak to prove property (T) for $\operatorname{SAut}(F_n)$ and $\operatorname{SL}_n(\mathbb{Z})$. In this paper, we adapt this technique to Laplacians in degree one. This allows to provide a lower bound for the cohomological Laplacian in degree one for $\operatorname{SL}_n(\mathbb{Z})$ for every unitary representation. In particular, one gets in that way an alternative proof of property (T) for $\operatorname{SL}_n(\mathbb{Z})$ whenever $n\geq 3$. |
| title | Inducing spectral gaps for the cohomological Laplacians of $\operatorname{SL}_n(\mathbb{Z})$ and $\operatorname{SAut}(F_n)$ |
| topic | Group Theory 20G10 |
| url | https://arxiv.org/abs/2404.10287 |