On symmetries of hyperbolic lattices of large rank

TL;DR

This paper proves that hyperbolic lattices of rank at least 46 have trivial exceptional lattices, implying they admit symmetries of maximal Salem degree, advancing understanding of their symmetry structures.

Contribution

It establishes a rank threshold (46) beyond which hyperbolic lattices have trivial exceptional lattices, revealing new symmetry properties of high-rank lattices.

Findings

Hyperbolic lattices of rank ≥ 46 have trivial exceptional lattices.

Such lattices admit symmetries of maximal Salem degree.

The result generalizes previous symmetry classifications.

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.