On products of isometries of hyperbolic space
type: Item
instance of: scholarly article
title: On products of isometries of hyperbolic space
zbMATH Open document ID: 1173.22008
author: Elisha Falbel
author: Richard A. Wentworth
published in: Topology and its Applications
publication date: 2009-08-20T00:00:00Z
review text: Let \({\mathbb F} = {\mathbb R}\), \({\mathbb C}\), or \({\mathbb Q}\), the real, complex, or quaternionic fields, and let \(PU(2, 1,{\mathbb F})\) denote the isometry group of the two-dimensional hyperbolic space over \({\mathbb F}\). The authors prove that: If \(C_1,\dots,C_k\), \(k\geq 3\), are arbitrary conjugacy classes of loxodromic elements of \(PU(2, 1,{\mathbb F})\), then (i) there exist \(g_1, \dots, g_k \in PU(2, 1, {\mathbb F})\), \(g_i \in C_i\), such that \(g_1\cdots g_k = I\) and (ii) the set \(\{(g_1, g_2, g_3): g_i\in C_i\), \(g_1 g_2 g_3 = I \}\) is compact modulo the diagonal action of conjugation by \(PU(2, 1,{\mathbb F})\). (i) could not be proved for the isometry group of higher-dimensional hyperbolic space.
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 51F15
zbMATH DE Number: 5597570
zbMATH Keywords: hyperbolic space
zbMATH Keywords: products of isometries
zbMATH Keywords: conjugacy classes
reviewed by: Victor V. Pambuccian
MaRDI profile type: Publication
full work available at URL: j.topol.2009.05.013
OpenAlex ID: W1982179304
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DOI: 10.1016/J.TOPOL.2009.05.013
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description: scientific article; zbMATH DE number 5597570