Cusp cross-section phenomena for arithmetic hyperbolic manifolds
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911240393064448 |
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| author | McCoy, Duncan Sell, Connor |
| author_facet | McCoy, Duncan Sell, Connor |
| contents | Although every flat manifold occurs as a cusp cross-section in at least one commensurability class of arithmetic hyperbolic manifolds, it turns out that some flat manifolds have the property that they occur as cusp cross-sections in precisely one commensurability class of arithmetic hyperbolic manifolds -- a phenomena which we will refer to as the UCC property. We construct flat manifolds with the UCC property in all dimensions $ n \geq 32 $. We also show that the number of distinct commensurability classes containing cusp cross-sections with the UCC property is unbounded. We also exhibit pairs of manifolds in all dimensions $ n \geq 24 $ that cannot arise as cusp cross-sections in the same commensurability class of arithmetic hyperbolic manifolds.
The main tool is previous work of the authors algebraically characterizing when a given flat manifold arises as the cusp cross-section of a manifold in a given commensurability class of arithmetic hyperbolic manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26127 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cusp cross-section phenomena for arithmetic hyperbolic manifolds McCoy, Duncan Sell, Connor Geometric Topology Although every flat manifold occurs as a cusp cross-section in at least one commensurability class of arithmetic hyperbolic manifolds, it turns out that some flat manifolds have the property that they occur as cusp cross-sections in precisely one commensurability class of arithmetic hyperbolic manifolds -- a phenomena which we will refer to as the UCC property. We construct flat manifolds with the UCC property in all dimensions $ n \geq 32 $. We also show that the number of distinct commensurability classes containing cusp cross-sections with the UCC property is unbounded. We also exhibit pairs of manifolds in all dimensions $ n \geq 24 $ that cannot arise as cusp cross-sections in the same commensurability class of arithmetic hyperbolic manifolds. The main tool is previous work of the authors algebraically characterizing when a given flat manifold arises as the cusp cross-section of a manifold in a given commensurability class of arithmetic hyperbolic manifolds. |
| title | Cusp cross-section phenomena for arithmetic hyperbolic manifolds |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2510.26127 |