The classification of Rank 3 Reflective Hyperbolic Lattices over Z[\sqrt{2}]

TL;DR

This paper classifies certain rank 3 hyperbolic lattices over Z[√2], adapting existing methods to identify those generated by reflections, and introduces new computational techniques to handle complex cases efficiently.

Contribution

It extends classification methods for hyperbolic lattices to the real quadratic setting and develops alternative algorithms to Vinberg's for computational efficiency.

Findings

Identified 432 strongly squarefree symmetric bilinear forms over Z[√2]

Produced a finite list of relevant quadratic forms

Developed new algorithms to replace Vinberg's for faster computation

Abstract

There are 432 strongly squarefree symmetric bilinear forms of signature $(2,1)$ defined over $\Z[\sqrt{2}]$ whose integral isometry groups are generated up to finite index by finitely many reflections. We adapted Allcock's method (based on Nikulin's) of analysis for the $2$-dimensional Weyl chamber to the real quadratic setting, and used it to produce a finite list of quadratic forms which contains all of the ones of interest to us as a sub-list. The standard method for determining whether a hyperbolic reflection group is generated up to finite index by reflections is an algorithm of Vinberg. However, for a large number of our quadratic forms the computation time required by Vinberg's algorithm was too long. We invented some alternatives, which we present here.