The reflective Lorentzian lattices of rank 3

TL;DR

This paper classifies all rank 3 Lorentzian lattices with reflection-generated isometry groups, extending previous work and employing computational and geometric techniques to analyze their Weyl chambers.

Contribution

It provides a comprehensive classification of symmetric bilinear forms of signature (2,1) with reflection-generated isometry groups, including enumeration and correction of prior definitions.

Findings

Classified 8595 lattices up to scale

Identified 374 distinct Weyl groups in 39 classes

Extended Nikulin's enumeration to non-square-free cases

Abstract

We classify all the symmetric integer bilinear forms of signature (2,1) whose isometry groups are generated up to finite index by reflections. There are 8595 of them up to scale, whose 374 distinct Weyl groups fall into 39 commensurability classes. This extends Nikulin's enumeration of the strongly square-free cases. Our technique is an analysis of the shape of the Weyl chamber, followed by computer work using Vinberg's algorithm and our "method of bijections". We also correct a minor error in Conway and Sloane's definition of their canonical 2-adic symbol.