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We give a version of Shimizu’s lemma for groups of complex hyperbolic isometries one of whose generators is a parabolic screw motion. Suppose that G is a discrete group containing a parabolic screw motion A and let B be any element of G not fixing the fixed point of A. Our result gives a bound on the radius of the isometric spheres of B and B−1 in terms of… (More)
A complex hyperbolic triangle group is a group generated by three complex reflections fixing complex slices (complex geodesics) in complex hyperbolic space. Our purpose in this paper is to improve the result in  and to discuss discreteness of complex hyperbolic triangle groups of type (n, n,∞; k).
We first demonstrate a family of isomorphisms between complex hyperbolic triangle groups and outline a systematic approach classifying the groups. Then we describe conditions that determine the discreteness of certain groups, in particular we prove a slightly weaker version of a conjecture given by Schwartz. Finally we collect together a list of known… (More)