Reflection length at infinity in hyperbolic reflection groups
In a discrete group generated by hyperplane reflections in the 𝑛-dimensional hyperbolic space, the reflection length of an element is the minimal number of hyperplane reflections in the group that suffices to factor the element. For a Coxeter group that arises in this way and does not split into a d...
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| Main Author: | |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
26. Juli 2024
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| In: |
Expert review of medical devices
Year: 2024, Pages: 1-23 |
| ISSN: | 1745-2422 |
| DOI: | 10.1515/jgth-2023-0073 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1515/jgth-2023-0073 Verlag, lizenzpflichtig, Volltext: https://www.degruyterbrill.com/document/doi/10.1515/jgth-2023-0073/html 
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| Author Notes: | Marco Lotz |
| Summary: | In a discrete group generated by hyperplane reflections in the 𝑛-dimensional hyperbolic space, the reflection length of an element is the minimal number of hyperplane reflections in the group that suffices to factor the element. For a Coxeter group that arises in this way and does not split into a direct product of spherical and affine reflection groups, the reflection length is unbounded. The action of the Coxeter group induces a tessellation of the hyperbolic space. After fixing a fundamental domain, there exists a bijection between the tiles and the group elements. We describe certain points in the visual boundary of the 𝑛-dimensional hyperbolic space for which every neighbourhood contains tiles of every reflection length. To prove this, we show that two disjoint hyperplanes in the 𝑛-dimensional hyperbolic space without common boundary points have a unique common perpendicular. |
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| Item Description: | Gesehen am 28.11.2024 |
| Physical Description: | Online Resource |
| ISSN: | 1745-2422 |
| DOI: | 10.1515/jgth-2023-0073 |