ON DISCRETE SUBGROUPS OF PU(1, 2; C) WITH HEISENBERG TRANSLATIONS

Abstract

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].