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

TL;DR

This paper investigates the conditions under which complex hyperbolic triangle groups of a specific type are discrete, identifying parameter ranges and possible orders of generating reflections, advancing understanding of their geometric properties.

Contribution

It classifies the discreteness of complex hyperbolic triangle groups of type $[m,m,0; n_1, n_2, 2]$ and determines the parameter intervals for discreteness and non-discreteness.

Findings

Identified possible orders for $n_1$ and $n_2$

Mapped parameter intervals for group discreteness

Established criteria for discreteness of these groups

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.