# Complex Hyperbolic Triangle Groups of Type $[m,m,0;3,3,2]$

## TL;DR

This paper investigates the conditions under which complex hyperbolic triangle groups of a specific type are discrete, by analyzing their parameter space and identifying intervals of discreteness and non-discreteness.

## Contribution

It provides a detailed analysis of the discreteness criteria for complex hyperbolic triangle groups of type [m,m,0;3,3,2], including the characterization of their parameter space.

## Key findings

- Identified intervals in parameter space corresponding to discrete groups.
- Established criteria distinguishing discrete from non-discrete groups.
- Mapped the parameter space for complex hyperbolic triangle groups of the specified type.

## Abstract

In this paper we study discreteness of complex hyperbolic triangle groups of type $[m,m,0;3,3,2]$, i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders $3,3,2$ 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 intervals in this parameter space that correspond to discrete and to non-discrete triangle groups.