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Discreteness of the complex hyperbolic ultra-parallel triangle groups

Wei Liao, Baohua Xie

Abstract

We prove that a family of complex hyperbolic ultra-parallel $[m_1, m_2, m_3]$-triangle group representations, where \( m_3 > 0 \), is discrete and faithful if and only if the isometry \( R_1(R_2R_1)^nR_3 \) is non-elliptic for some positive integer \( n \). Additionally, we investigate the special case where \( m_3 = 0 \) and provide a substantial improvement upon the main result by Monaghan, Parker, and Pratoussevitch.

Discreteness of the complex hyperbolic ultra-parallel triangle groups

Abstract

We prove that a family of complex hyperbolic ultra-parallel -triangle group representations, where , is discrete and faithful if and only if the isometry \( R_1(R_2R_1)^nR_3 \) is non-elliptic for some positive integer . Additionally, we investigate the special case where and provide a substantial improvement upon the main result by Monaghan, Parker, and Pratoussevitch.

Paper Structure

This paper contains 15 sections, 26 theorems, 152 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background
  3. The unit ball model of complex hyperbolic space
  4. Complex hyperbolic ultra-parallel triangle groups
  5. Bisectors
  6. The Siegel domain model of complex hyperbolic space
  7. A discreteness criterion
  8. Parameterization
  9. Discreteness result
  10. Outline of the Proof
  11. Proof of Proposition \ref{['proposition']}
  12. Technical Lemmas
  13. An Algorithm
  14. A New Family of Complex Hyperbolic $[m_1, m_2, m_3]$-Triangle Group Representations with $m_3 > 0$
  15. Discreteness of the complex hyperbolic ultra-parallel $[m_1, m_2, 0]$-triangle group representations

Key Result

Theorem 1.1

A complex hyperbolic ultra-parallel $[m_1, m_2, m_3]$-triangle group representation with angular invariant $\alpha$ (see Proposition angular for the precise definition) and $m_3 > 0$ is discrete and faithful if the following three conditions are satisfied: Here $r_j=\cosh(\frac{m_j}{2})$ for $j=1, 2, 3$.

Figures (7)

  • Figure 1: As an example, consider $r_3 = 1.01$ and $j_0 = 3$. In each set $K_j'$ for $j = 1, 2, 3$, the corresponding complex hyperbolic ultra-parallel $[m_1, m_2, m_3]$-triangle group representations are discrete and faithful if and only if the element $R_1(R_2R_1)^j R_3$ is non-elliptic.
  • Figure 2: In each region $\mathcal{K}_n$, the red area corresponds to complex hyperbolic ultra-parallel $[m_1, m_2, 0]$-triangle group representations. These representations are discrete and faithful if and only if the element $w^{(n)} = R_1 (R_2 R_1)^n R_3$ is non-elliptic. Furthermore, in each region $\mathcal{K}_n$, the entire red region corresponds to the result in Theorem \ref{['main thm 3']}, while the area below the blue curve corresponds to the result in Proposition 1 of mpp.
  • Figure 3:
  • Figure 4: Schematic diagram for $p_3^0 = \Pi_{(-1, 1)}(p_3)$ in $D_3$.
  • Figure 5: Schematic diagram for $p_3^0 = \Pi_{(-1, 1)}(p_3)$ in $D_4$.
  • ...and 2 more figures

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 35 more