Classical projective geometry and arithmetic groups
type: Item
instance of: scholarly article
title: Classical projective geometry and arithmetic groups
zbMATH Open document ID: 0732.20029
DOI: 10.1007/BF01459253
author: Mark McConnell
published in: Mathematische Annalen
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164826
review text: We construct cell complexes which are homotopic to certain locally symmetric spaces. Each cell complex is the geometric realization of a partially ordered set \({\mathcal C}\); the elements of \({\mathcal C}\) are configurations in projective space. The locally symmetric spaces we study are modular varieties \(K\setminus G/\Gamma\), with \(G=SL(n,{\mathbb{R}})\), \(n=2,3,4\), and \(\Gamma =\Gamma (p)\subset SL(n,{\mathbb{Z}})\) a principal congruence subgroup (p is an odd prime). Our cell complex is embedded in \(K\setminus G/\Gamma\) as a deformation retract. The deformation retract itself was found by Soulé and Lannes and studied by Ash. We generalize their work and include two main points of innovation. First, we have explicit results for the specific \(\Gamma\) we study. Second, we introduce the projective configurations in \({\mathcal C}\) as objects of study in their own right. As a by-product of this work we prove that the Soulé-Lannes retract is dual to the first Voronoi decomposition of \(K\setminus G\) for all n.
Mathematics Subject Classification ID: 20H05
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 11F06
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 14L35
Mathematics Subject Classification ID: 14J20
zbMATH DE Number: 4210417
zbMATH Keywords: cell complexes
zbMATH Keywords: locally symmetric spaces
zbMATH Keywords: configurations
zbMATH Keywords: projective space
zbMATH Keywords: modular varieties
zbMATH Keywords: principal congruence subgroup
zbMATH Keywords: deformation retract
zbMATH Keywords: projective configurations
zbMATH Keywords: Voronoi decomposition
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4210417