Reflective anisotropic hyperbolic lattices of rank $4$

TL;DR

This paper provides a complete classification of 1.2-reflective maximal anisotropic hyperbolic lattices of rank 4, advancing understanding of their automorphism groups and reflection properties.

Contribution

It offers the first comprehensive classification of 1.2-reflective maximal anisotropic hyperbolic lattices of rank 4, a previously unresolved problem.

Findings

Classified all 1.2-reflective maximal anisotropic lattices of rank 4

Identified finite index subgroups generated by 1- and 2-reflections

Enhanced understanding of automorphism groups of hyperbolic lattices

Abstract

A hyperbolic lattice is called \textit{$1.2$-reflective} if the subgroup of its automorphism group generated by all $1$- and $2$-reflections is of finite index. The main result of this article is a complete classification of $1.2$-reflective maximal anisotropic lattices of rank $4$.