The classification of rank 3 reflective hyperbolic lattices over

Abstract

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

Reflective Lorentzian lattices of signature (5,1)

“…[All12]). The transition to number fields was studied in [Mar15] where rank 3 reflective lattices over Z[ √ 2] were classified.…”

Arithmetic hyperbolic reflection groups

“…In her PhD thesis [Mar15b], Mark studied the classification of rank 3 reflective lattices over quadratic extensions of Q. To this end she adapted the method of Allcock to the real quadratic setting and used a different algorithm for checking reflectivity of a quadratic form termed the walking algorithm.…”

“…The latter condition implies that all the hyperplanes Π e i are essential. Note that if k = Q, its integers do not form a discrete subset of R. Bugaenko showed that regardless of this, the arithmeticity assumption implies that the set of distances considered above is discrete and hence we can always choose the smallest one (see [Bug84], [Bug90], [Bug92] or [Mar15b]). The algorithm terminates if it generates a configuration P = i Π − e i that has finite volume, in which case the form f is reflective.…”

From geometry to arithmetic of compact hyperbolic Coxeter polytopes

“…In this case we are given bounds on all elements of the matrix G(u 1 , u 2 , u 3 , u 4 ), because all the faces F i are pairwise intersecting, excepting, possibly, the pair of faces F 3 and F 4 . But if they do not intersect, then the distance between these faces is bounded by inequality (16). Thus, all entries of the matrix G(u 1 , u 2 , u 3 , u 4 ) are integer and bounded, so there are only finitely many possible matrices G(u 1 , u 2 , u 3 , u 4 ).…”

“…To reduce the enumeration of matrices G(u 1 , u 2 , u 3 , u 4 ) one can use Main Theorem 1 and Theorem 2.2 enabling us to get much sharper bounds on the number |(u 3 , u 4 )| than in inequality (16).…”