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arXiv · 2602.05652

On symmetries of hyperbolic lattices of large rank

Torben Grabbel·Gebhard Martin·Giacomo Mezzedimi·Maia Raitz von Frentz·Paul Jakob Schmidt
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Abstract

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.

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Torben Grabbel, Gebhard Martin, Giacomo Mezzedimi, Maia Raitz von Frentz, Paul Jakob Schmidt. 2026-02-05. On symmetries of hyperbolic lattices of large rank. https://arxiv.org/abs/2602.05652

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