Maximal Discrete Subgroups of $SO^+(2,n+2)$

TL;DR

This paper characterizes the maximal discrete subgroups of the special orthogonal group $SO^+(2,n+2)$ that contain the discriminant kernel of certain even lattices, providing explicit descriptions and examples involving root lattices.

Contribution

It provides a complete characterization of maximal discrete subgroups of $SO^+(2,n+2)$ containing specific lattice kernels, including their relation to normalizers and integral matrices.

Findings

Maximal discrete subgroups coincide with normalizers in $SO^+(2,n+2)$.

Explicit descriptions for lattices containing two hyperbolic planes.

Examples involving irreducible root lattices.

Abstract

We characterize the maximal discrete subgroups of $SO^+(2,n+2)$, which contain the discriminant kernel of an even lattice, which contains two hyperbolic planes over $\mathbb{Z}$. They coincide with the normalizers in $SO^+(2,n+2)$ and are given by the group of all integral matrices inside $SO^+(2,n+2)$, whenever the underlying lattice is maximal even. Finally we deal with the irreducible root lattices as examples.