The classification of rank 3 reflective hyperbolic lattices over Z([square root of 2])



Journal Title

Journal ISSN

Volume Title



There are 432 strongly squarefree symmetric bilinear forms of signature (2,1) defined over Z([square root of 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.