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

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.