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Complex Hyperbolic Triangle Groups of Type $[m,m,0; n_1, n_2, 2]$

Sam Povall, Anna Pratoussevitch

Abstract

In this paper we study the discreteness of complex hyperbolic triangle groups of type $[m,m,0; n_1, n_2, 2]$, i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders $n_1, n_2, 2$ in complex geodesics with pairwise distances $m, m, 0$. For fixed $m$, the parameter space of such groups is of real dimension one. We determine the possible orders for $n_1$ and $n_2$ and also intervals in the parameter space that correspond to discrete and non-discrete triangle groups.

Complex Hyperbolic Triangle Groups of Type $[m,m,0; n_1, n_2, 2]$

Abstract

In this paper we study the discreteness of complex hyperbolic triangle groups of type , i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders in complex geodesics with pairwise distances . For fixed , the parameter space of such groups is of real dimension one. We determine the possible orders for and and also intervals in the parameter space that correspond to discrete and non-discrete triangle groups.

Paper Structure

This paper contains 22 sections, 13 theorems, 101 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Complex hyperbolic plane:
  4. Complex geodesics:
  5. Complex reflections:
  6. Complex hyperbolic triangle groups:
  7. Angular invariant:
  8. Heisenberg group:
  9. Chains:
  10. Heisenberg isometries:
  11. Products of reflections in chains:
  12. Bisectors and spinal spheres:
  13. Parametrisation of complex hyperbolic triangle groups of type $[m_1, m_2, 0;n_1,n_2,n_3]$
  14. Orders of Reflection
  15. Compression Property
  16. ...and 7 more sections

Key Result

Theorem 1.1

An ultra-parallel complex hyperbolic triangle group of type $[m_1,m_2,0;n_1,n_2,n_3]$ can only be discrete if the unordered pair of orders of the complex reflections $\iota_1$ and $\iota_2$ is one of

Figures (2)

  • Figure 1: Discreteness and non-discreteness results in the $(m,{\alpha})$-space.
  • Figure 2: Chains $C_1$, $C_2$ and $C_3$ (figure from andyanna).

Theorems & Definitions (24)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Remark 1.4
  • Proposition 2.1
  • Proposition 4.1
  • proof
  • Theorem 4.2
  • Proposition 4.3
  • Remark 4.4
  • ...and 14 more