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

## TL;DR

This paper classifies certain four-dimensional hyperbolic lattices that are generated by simple reflections, revealing geometric properties of their fundamental domains in hyperbolic space.

## Contribution

It provides the first classification of $(1,2)$-reflective anisotropic hyperbolic lattices of rank 4, based on geometric analysis of their fundamental polyhedra.

## Key findings

- Fundamental polyhedron contains an edge with small distance between framing faces.
- Classification of $(1,2)$-reflective anisotropic hyperbolic lattices of rank 4 achieved.
- Geometric properties of reflection groups in hyperbolic space elucidated.

## Abstract

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 $4$.