On discrete subgroups of PU(1, 2;C) with Heisenberg translations

In the study of discrete groups it is important to find conditions for a group to be discrete. Given a discrete subgroup of Mo $ bius transformations containing a parabolic element with fixed point ¢, a classical result, called Shimizu’s lemma, gives a uniform bound on the radii of isometric circles of those elements of the group not fixing ¢