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In this paper, we consider ultra‐parallel complex hyperbolic triangle groups of type    [   m  1   ,   m  2   ,  0  ]   , that is, 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.

Discreteness of ultra‐parallel complex hyperbolic triangle groups of type [ m 1 , m 2 , 0 ]