Uniform discreteness and Heisenberg translations
type: Item
instance of: scholarly article
title: Uniform discreteness and Heisenberg translations
zbMATH Open document ID: 0881.32013
DOI: 10.1007/PL00004315
author: John R. Parker
published in: Mathematische Zeitschrift
publication date: 1997-09-04T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 57S30
zbMATH DE Number: 1057302
zbMATH Keywords: projective unitary group
zbMATH Keywords: holomorphic isometries
zbMATH Keywords: Heisenberg translation
zbMATH Keywords: complex hyperbolic space
MaRDI profile type: Publication
full work available at URL: pl00004315
OpenAlex ID: W2070018599
description: scientific article; zbMATH DE number 1057302