Hyperbolic 3-manifolds with uniform spectral gap for coclosed 1-forms
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909185118044160 |
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| author | Abdurrahman, Amina Adve, Anshul Giri, Vikram Lowe, Ben Zung, Jonathan |
| author_facet | Abdurrahman, Amina Adve, Anshul Giri, Vikram Lowe, Ben Zung, Jonathan |
| contents | We study two quantifications of being a homology sphere for hyperbolic 3-manifolds, one geometric and one topological: the spectral gap for the Laplacian on coclosed 1-forms and the size of the first torsion homology group. We first construct a sequence of closed hyperbolic integer homology spheres with volume tending to infinity and a uniform coclosed 1-form spectral gap. This answers a question asked by Lin--Lipnowski. We also find sequences of hyperbolic rational homology spheres with the same properties that geometrically converge to a tame limit manifold. Moreover, we show that any such sequence must have unbounded torsion homology growth. Finally we show that a sequence of closed hyperbolic rational homology 3-spheres with uniformly bounded rank and a uniform coclosed 1-form spectral gap must have torsion homology that grows exponentially in volume. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19039 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hyperbolic 3-manifolds with uniform spectral gap for coclosed 1-forms Abdurrahman, Amina Adve, Anshul Giri, Vikram Lowe, Ben Zung, Jonathan Geometric Topology Differential Geometry Spectral Theory We study two quantifications of being a homology sphere for hyperbolic 3-manifolds, one geometric and one topological: the spectral gap for the Laplacian on coclosed 1-forms and the size of the first torsion homology group. We first construct a sequence of closed hyperbolic integer homology spheres with volume tending to infinity and a uniform coclosed 1-form spectral gap. This answers a question asked by Lin--Lipnowski. We also find sequences of hyperbolic rational homology spheres with the same properties that geometrically converge to a tame limit manifold. Moreover, we show that any such sequence must have unbounded torsion homology growth. Finally we show that a sequence of closed hyperbolic rational homology 3-spheres with uniformly bounded rank and a uniform coclosed 1-form spectral gap must have torsion homology that grows exponentially in volume. |
| title | Hyperbolic 3-manifolds with uniform spectral gap for coclosed 1-forms |
| topic | Geometric Topology Differential Geometry Spectral Theory |
| url | https://arxiv.org/abs/2404.19039 |