On symmetries of hyperbolic lattices of large rank

TL;DR

Let $L$ be an even, integral hyperbolic lattice and $\mathrm{Aut}(\mathcal{D}_L)=\mathrm{O}^+(L)/W(L)$ its symmetry group. The paper proves that if $\mathrm{rk}(L)\ge 46$, then the exceptional lattice $E(L)$, generated by vectors with finite orbit under $\mathrm{Aut}(\mathcal{D}_L)$, is trivial, which in turn guarantees the existence of a symmetry of maximal Salem degree. The strategy reduces to overlattices of the form $L'\cong U\oplus E_8^n\oplus M$ with $n\ge 5$ and $\mathrm{rk}(M)\le 11$, and uses the isometry $U\oplus E_8^3\simeq U\oplus \Lambda$ (the Leech lattice) to show $E(L)\subseteq M$; a finite root analysis and cusps with infinite stabilizers then force $E(L')=0$, hence $E(L)=0$. These results yield finiteness consequences for hyperbolic lattices with non-trivial exceptional lattices and illuminate Salem-degree phenomena in automorphism groups of K3 surfaces, via their Picard lattices and elliptic fibrations.

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.

Figures (1)

• Figure 1: Highest roots of ADE lattices

Theorems & Definitions (30)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3: Yu
• Corollary 1.4
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof