On symmetries of hyperbolic lattices of large rank
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910012925804544 |
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| author | Grabbel, Torben Martin, Gebhard Mezzedimi, Giacomo von Frentz, Maia Raitz Schmidt, Paul Jakob |
| author_facet | Grabbel, Torben Martin, Gebhard Mezzedimi, Giacomo von Frentz, Maia Raitz Schmidt, Paul Jakob |
| contents | For an even, integral hyperbolic lattice $L$, the symmetry group of $L$ is the quotient of the group of isometries of $L$ by the Weyl subgroup of $(-2)$-reflections. Following Nikulin, the exceptional lattice of $L$ is defined as the sublattice generated by elements that have finite orbit under the symmetry group of $L$. We prove that every hyperbolic lattice of rank at least $46$ has trivial exceptional lattice. In particular, every such lattice admits a symmetry of maximal Salem degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05652 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On symmetries of hyperbolic lattices of large rank Grabbel, Torben Martin, Gebhard Mezzedimi, Giacomo von Frentz, Maia Raitz Schmidt, Paul Jakob Number Theory Algebraic Geometry 11H56, 14J50, 51M10 For an even, integral hyperbolic lattice $L$, the symmetry group of $L$ is the quotient of the group of isometries of $L$ by the Weyl subgroup of $(-2)$-reflections. Following Nikulin, the exceptional lattice of $L$ is defined as the sublattice generated by elements that have finite orbit under the symmetry group of $L$. We prove that every hyperbolic lattice of rank at least $46$ has trivial exceptional lattice. In particular, every such lattice admits a symmetry of maximal Salem degree. |
| title | On symmetries of hyperbolic lattices of large rank |
| topic | Number Theory Algebraic Geometry 11H56, 14J50, 51M10 |
| url | https://arxiv.org/abs/2602.05652 |