Uniform discreteness and Heisenberg screw motions

We give aversion of Shimizu’s lemma for groups of complex hyper-bolic isometries one of whose generators is aHeisenberg screw motion. Our main result interpolates between known results for groups with a generator that is avertical translation or aHeisenberg rotation. We also give an interpretation of our result in terms of the relation be-tween radii of isometric spheres and their distance from the axis of the Heisenberg screw motion. 1Introduction Shimizu’s lemma [12] gives anecessary condition for asubgroup of PSL(2, R) containing aparabolic element fixing $\infty $ to be discrete. It was generalised for discrete groups of higher dimensional real hyperbolic isometries contain-ing aparabolic element by Leutbecher [8], Wielenberg [14], Ohtake [9] and Waterman [13]. The hyperbolic plane is not only real hyperbolic 2-space $\mathrm{H}_{\mathbb{R}}^{2} $ , but also complex hyperbolic 1-space $\mathrm{H}_{\mathbb{C}}^{1} $. Therefore it is natural to generalise Shimizu’s lemma to discrete groups of isometries of higher dimen-sional complex hyperbolic $\mathrm{H}_{\mathbb{C}}^{n} $ space containing aparabolic element. This is part of awider project to give generalisations of Jorgensen’s inequali