Reflection groups on the octave hyperbolic plane
type: Item
instance of: scholarly article
title: Reflection groups on the octave hyperbolic plane
zbMATH Open document ID: 0932.51002
author: Daniel Allcock
published in: Journal of Algebra
publication date: 1999-03-30T00:00:00Z
review text: The paper presents three examples of discrete reflection groups. Two of them act with finite covolume on the octave hyperbolic plane \(\mathbb{Q} H^2\), the third one is a discrete group of isometries of the octave hyperbolic line \(\mathbb{O} H^1\). After having introduced the algebra \(\mathbb{Q}\) of octaves the author defines the octave projective plane \(\mathbb{Q} P^2\) and the octave hyperbolic plane \(\mathbb{Q} H^2\) by using the Jordan algebra approach of \textit{H. Freudenthal} [Geom. Dedicata 19, 7-63 (1985; Zbl 0573.51004)]. He considers two reflections \(R\), \(R'\). The discrete groups of isometries of the plane \(\mathbb{Q} H^2\) will be generated by conjugates of \(R\) \(R'\). They are subgroups of the Lie group \(\Aut\mathbb{Q} H^2\). The discrete subgroup of isometries of \(\mathbb{Q} H^1\) is constructed by using the fact that \(\mathbb{Q} H^1\) is isomorphic to the real hyperbolic space \(H^8\).
reviewed by: Wolfgang Nolte
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 51H99
Mathematics Subject Classification ID: 17C50
zbMATH DE Number: 1270861
zbMATH Keywords: discrete reflection groups
zbMATH Keywords: octave hyperbolic plane
MaRDI profile type: Publication
full work available at URL: jabr.1998.7671
OpenAlex ID: W1980391800
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DOI: 10.1006/JABR.1998.7671
description: scientific article; zbMATH DE number 1270861