Thin hyperbolic reflection groups

TL;DR

This paper classifies and describes thin hyperbolic reflection groups, showing they are enumerable and related to Vinberg's algorithm applied to Lorentzian lattices, expanding understanding of discrete groups in hyperbolic geometry.

Contribution

It provides a comprehensive description of all thin hyperbolic reflection groups and links their construction to Vinberg's algorithm, regardless of lattice reflectivity.

Findings

All thin hyperbolic reflection groups are enumerable.

Vinberg's algorithm applied to non-reflective lattices produces infinite thin reflection groups.

Every thin hyperbolic reflection group is a subgroup of a group from Vinberg's algorithm.

Abstract

We study a family of Zariski dense finitely generated discrete subgroups of $\mathrm{Isom}(\mathbb{H}^d)$, $d \geqslant 2$, defined by the following property: any group in this family contains at least one reflection in a hyperplane. As an application we obtain a general description of all thin hyperbolic reflection groups. In particular, we show that the Vinberg algorithm applied to a non-reflective Lorentzian lattice gives rise to an infinite sequence of thin reflection subgroups in $\mathrm{Isom}(\mathbb{H}^d)$, for any $d \geqslant 2$. Moreover, every such group is a subgroup of a group produced by the Vinberg algorithm applied to a Lorentzian lattice independently on the latter being reflective. As a consequence, all thin hyperbolic reflection groups are enumerable.