In the study of discrete groups it is important to find conditions for a group to be
discrete. Given a discrete subgroup of Möbius 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 ∞.
Recently Parker [8] has shown that if a discrete subgroup G of PU(1, n; C) contains
a Heisenberg translation g, then any element of G not sharing a fixed point with g has
an isometric sphere whose radius is bounded above by a function of the translation
length of g at its centres. Parker's theorem is considered as a generalization of
Shimizu's lemma. In [1] Basmajian and Miner have independently obtained
qualitatively similar results for discrete subgroups of PU(1, 2; C) by using their stable
basin theorem.
The purpose of this paper is twofold. First we improve the stable basin theorem,
and second we show that under some conditions Parker's theorem yields the
discreteness conditions of Basmajian and Miner for groups with a Heisenberg
translation. The latter answers a question posed in [8].