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Thin hyperbolic reflection groups

Nikolay Bogachev, Alexander Kolpakov

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.

Thin hyperbolic reflection groups

Abstract

We study a family of Zariski dense finitely generated discrete subgroups of , , 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 , for any . 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.
Paper Structure (10 sections, 11 theorems, 5 equations, 1 figure)

This paper contains 10 sections, 11 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lorentzian lattices and hyperbolic reflection groups
  4. Zariski dense discrete subgroups of $\mathrm{Isom}(\mathbb{H}^d)$
  5. Algebraic proof of Theorem \ref{['th:main']}
  6. Geometric proof of Theorem \ref{['th:main']}
  7. Proof of Theorem \ref{['th:thin-2']}
  8. Examples
  9. Non--reflective lattice with roots
  10. A lattice without roots

Key Result

Theorem 1.1

Let $\Gamma < \mathrm{Isom}(\mathbb{H}^d)$ be a finitely generated Zariski dense discrete group containing at least one reflection. Then $\Gamma$ contains a discrete Zariski dense subgroup generated by finitely many reflections. Moreover, if the maximal reflection subgroup of $\Gamma$ is infinite--i

Figures (1)

  • Figure 1: The Coxeter scheme for $v_1, \ldots, v_4$ (Section \ref{['non-ref1']})

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1: È. B. Vinberg, Vin72
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 12 more