Simple Tilings of Nilpotent Lie Groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918398593597440 |
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| author | Hansen, Kyle |
| author_facet | Hansen, Kyle |
| contents | We define simple tilings in the general context of a $G$-tiling on a Riemannian homogeneous space $M$ to be tilings by Riemannian simplices. As evidence that this definition is natural, we prove that a large class of tilings of $M$ are MLD to simple ones. We demonstrate the utility of this definition by generalizing previously known results about simple tilings of Euclidean space. In particular, it is shown that a simple tiling space of a rational, connected, simply connected, nilpotent Lie group is homeomorphic to a rational tiling space, that is, a tiling space for which displacement between vertices take on rational values. Hence, such a tiling space is a fiber bundle over a nilmanifold. We further sketch a proof of the fact that there is an isomorphism between Čech cohomology and pattern equivariant cohomology of simple tilings in connected, simply connected, nilpotent Lie groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10194 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simple Tilings of Nilpotent Lie Groups Hansen, Kyle Dynamical Systems Metric Geometry We define simple tilings in the general context of a $G$-tiling on a Riemannian homogeneous space $M$ to be tilings by Riemannian simplices. As evidence that this definition is natural, we prove that a large class of tilings of $M$ are MLD to simple ones. We demonstrate the utility of this definition by generalizing previously known results about simple tilings of Euclidean space. In particular, it is shown that a simple tiling space of a rational, connected, simply connected, nilpotent Lie group is homeomorphic to a rational tiling space, that is, a tiling space for which displacement between vertices take on rational values. Hence, such a tiling space is a fiber bundle over a nilmanifold. We further sketch a proof of the fact that there is an isomorphism between Čech cohomology and pattern equivariant cohomology of simple tilings in connected, simply connected, nilpotent Lie groups. |
| title | Simple Tilings of Nilpotent Lie Groups |
| topic | Dynamical Systems Metric Geometry |
| url | https://arxiv.org/abs/2402.10194 |