Uniform discreteness and Heisenberg translations
Let \(G\) be a discrete subgroup of \({\mathbf P} {\mathbf U} (n,1)\) containing a Heisenberg translation \(g\). The purpose of this paper is to show that 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 centre. This may be thought of as a complex hyperbolic version of the generalisation of Shimizu's lemma given by Waterman. We use this to show that if the subgroup of \(G\) stabilising the fixed point of \(g\) is a group of Heisenberg translations then \(G\) either leave a horoball or sub-horospherical region precisely invariant provided. Consequences involving the distance of generators of \(G\) from the identity and bounds on the largest embedded ball in the quotient of complex hyperbolic space by \(G\) are also given.
scientific article; zbMATH DE number 1057302