Discreteness of Ultra-Parallel Complex Hyperbolic Triangle Groups of Type $[m_1,m_2,0]$

TL;DR

This paper investigates the conditions under which certain complex hyperbolic triangle groups of type [m_1,m_2,0], generated by complex reflections, are discrete or not, revealing connections to elliptic elements.

Contribution

It provides new results on the discreteness and non-discreteness of ultra-parallel complex hyperbolic triangle groups of a specific type, linking these properties to elliptic elements.

Findings

Discreteness conditions for groups of type [m_1,m_2,0]

Non-discreteness criteria for these groups

Connection between discreteness and ellipticity of elements

Abstract

In this paper we consider ultra-parallel complex hyperbolic triangle groups of type $[m_1,m_2,0]$, i.e. groups of isometries of the complex hyperbolic plane, generated by complex reflections in three ultra-parallel complex geodesics two of which intersect on the boundary. We prove some discreteness and non-discreteness results for these groups and discuss the connection between the discreteness results and ellipticity of certain group elements.