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

Abstract

There are 432 strongly squarefree symmetric bilinear forms of signature $(2,1)$ defined over $\mathbb{Z}[\sqrt{2}]$ 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.

The first 40 pages or so are the paper. The rest of it is a giant table

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

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