Higher Kazhdan property and unitary cohomology of arithmetic groups
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866912882138021888 |
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| author | Bader, Uri Sauer, Roman |
| author_facet | Bader, Uri Sauer, Roman |
| contents | Notions of higher Kazhdan property can be defined in terms of vanishing of unitary group cohomology in higher degrees. Garland's theorem for simple groups over non-archimedean fields provides the first examples of a higher Kazhdan property. We prove a version of Garland's theorem for simple Lie groups and their lattices. We generalize theorems of Borel and Borel-Yang about the invariance of the cohomology of lattices in semisimple Lie groups and adelic groups by improving the stability range and allowing for arbitrary unitary representations as coefficients. A novelty of our approach is the use of methods from geometric group theory and -- in the case of rank 1 -- from Clozel's work on the spectral gap property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_06517 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Higher Kazhdan property and unitary cohomology of arithmetic groups Bader, Uri Sauer, Roman Representation Theory Algebraic Topology Group Theory 22E41 (Primary), 22E46, 20G10 (Secondary) Notions of higher Kazhdan property can be defined in terms of vanishing of unitary group cohomology in higher degrees. Garland's theorem for simple groups over non-archimedean fields provides the first examples of a higher Kazhdan property. We prove a version of Garland's theorem for simple Lie groups and their lattices. We generalize theorems of Borel and Borel-Yang about the invariance of the cohomology of lattices in semisimple Lie groups and adelic groups by improving the stability range and allowing for arbitrary unitary representations as coefficients. A novelty of our approach is the use of methods from geometric group theory and -- in the case of rank 1 -- from Clozel's work on the spectral gap property. |
| title | Higher Kazhdan property and unitary cohomology of arithmetic groups |
| topic | Representation Theory Algebraic Topology Group Theory 22E41 (Primary), 22E46, 20G10 (Secondary) |
| url | https://arxiv.org/abs/2308.06517 |