秩为3的反射洛伦兹晶格

Virtually special embeddings of integral Lorentzian lattices

The automorphism groups of integral Lorentzian lattices act by isometries on hyperbolic space with finite covolume. In the case of reflective integral lattices, the automorphism groups are commensurable to arithmetic hyperbolic reflection groups. However, for a fixed dimension, there is only finitely many reflective integral Lorentzian lattices, and these can only occur in small dimensions. The goal of this note is to construct embeddings of low-dimensional integral Lorentzian lattices into unimodular Lorentzian lattices associated to right-angled reflection groups. As an application, we const…

秩为3的反射格的完整表

Unabridged table of reflective lattices of rank 3

这是所有8595个三阶反射洛伦兹晶格的完整表格，旨在作为作者论文分类的补充。那篇论文中的节略表也是完整的，但不太明确，所以这篇论文的主要目的是存档。TeX源代码同时也是一个Perl脚本，它在运行时以计算机可读的格式打印出所有的网格。

秩为$4$的反射各向异性双曲格

Reflective anisotropic hyperbolic lattices of rank $4$

一个双曲格称为\textit{$1.2$-reflective}，如果由所有$1$ -和$2$ -反射生成的自同构群的子群是有限指数的。本文的主要结果是对秩为$4$的$1.2$ -反射极大各向异性格的完全分类。

Classification of $(1{,}2)$-reflective anisotropic hyperbolic lattices of rank $4$

A hyperbolic lattice is called \textit{$(1{,}2)$-reflective} if its automorphism group is generated by $1$- and $2$-reflections up to finite index. In this paper we prove that the fundamental polyhedron of a $\mathbb{Q}$-arithmetic cocompact reflection group in the three-dimensional Lobachevsky space contains an edge such that the distance between its framing faces is small enough. Using this fact we obtain a classification of $(1{,}2)$-reflective anisotropic hyperbolic lattices of rank $4$.

Z[ \sqrt{2} ]上秩3反射双曲格的分类

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

定义在$\Z[\sqrt{2}]$上的签名$(2,1)$的432个强无平方对称双线性型，它们的积分等距群是通过多次反射生成的直到有限指数.我们适应Allcock的方法（基于Nikulin的）的分析$2$维Weyl室的实二次设置，并使用它来产生一个有限的列表的二次型，其中包含所有的感兴趣的我们作为一个子列表。用于确定双曲反射群是否由反射生成到有限折射率的标准方法是Vinberg算法。然而，对于大量的我们的二次型的计算时间所需的Vinberg的算法太长。我们发明了一些替代品，我们在这里介绍。