Complex Hyperbolic Triangle Groups with 2-fold Symmetry

TL;DR

This paper classifies complex hyperbolic triangle groups with 2-fold symmetry, focusing on those with elliptic elements of finite order, and identifies four potential discrete groups.

Contribution

It provides a complete classification of 2-fold symmetric complex hyperbolic triangle groups that could be discrete, based on their elliptic elements and finite order conditions.

Findings

Identified four types of candidate discrete groups.

Focused on groups generated by complex reflections with specific angles.

Classified groups based on elliptic elements of finite order.

Abstract

In this paper we will consider the 2-fold symmetric complex hyperbolic triangle groups generated by three complex reflections through angle 2pi/p with p no smaller than 2. We will mainly concentrate on the groups where some elements are elliptic of finite order. Then we will classify all such groups which are candidates for being discrete. There are only 4 types.