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The classification of rank 3 reflective hyperbolic lattices over $\mathbb{Z}[\sqrt{2}]$

Published online by Cambridge University Press:  09 December 2016

ALICE MARK
ALICE MARK*
Affiliation:
School of Mathamatical and Statistical Sciences, Arizona State University, P.O. Box 871804, Tempe, AZ 85287-1804, U.S.A.
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 2 , March 2018 , pp. 221 - 257
Copyright
Copyright © Cambridge Philosophical Society 2016 

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