The reflective Lorentzian lattices of rank 3

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.

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