Discreteness criteria for subgroups in complex hyperbolic space
type: Item
instance of: scholarly article
title: Discreteness criteria for subgroups in complex hyperbolic space
zbMATH Open document ID: 1009.32016
author: Binlin Dai
author: Bing Nai
author: Ainong Fang
published in: Proceedings of the Japan Academy. Series A
publication date: 2002-08-18T00:00:00Z
review text: The authors study discreteness criteria for subgroups of \(U(1,n; \mathbb{C})\) in complex hyperbolic space \(H_n^\mathbb{C}\). The main result in this paper is as follows: A nonelementary subgroup \(G\) of \(U(1,n;\mathbb{C})\) in complex hyperbolic space \(H_n^\mathbb{C}\) with Condition A is discrete if and only if every two generator subgroup is discrete. Here we say that \(G\) satisfies Condition A provided that \(G\) has no sequence \(\{g_j\}\) of distinct elements of finite order such that \(\text{Card (fix}(g_j))= \infty\) in \(\partial H_n^\mathbb{C}\) and \(g_j\to I\) as \(j\to\infty\). They also prove that if a nonelementary subgroup \(G\) of \(U(1,n;\mathbb{C})\) contains a sequence \(\{g_j\}\) of distinct elements with \(\text{Card (fix}(g_j)\cap \partial H_n^\mathbb{C}) \neq\infty\) and \(g_j \to I\) as \(j\to\infty\), then \(G\) contains a non-discrete, nonelementary two generator subgroup.
reviewed by: Yoshihiro Aihara
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 32M99
Mathematics Subject Classification ID: 22E99
zbMATH DE Number: 1782975
zbMATH Keywords: limit set
zbMATH Keywords: elementary groups
zbMATH Keywords: discreteness
zbMATH Keywords: complex hyperbolic space
MaRDI profile type: Publication
full work available at URL: pjaa.77.168
OpenAlex ID: W2014452331
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DOI: 10.3792/PJAA.77.168
description: scientific article; zbMATH DE number 1782975