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

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 $4$.