Geometric and topological aspects of Coxeter groups and buildings
type: Item
instance of: scholarly article
title: Geometric and topological aspects of Coxeter groups and buildings
zbMATH Open document ID: 1415.20005
DOI: 10.4171/189
author: Y. Aharonov
published in: Zurich Lectures in Advanced Mathematics
publication date: 2018-06-20T00:00:00Z
review text: This book originated from lectures for graduate students given by the author at the ETH Zurich and the Institute for Mathematical Research (FIM). It has the liveliness and occasionally the informality of live lectures. This is a text directed to graduate students, who have already a broad knowledge of mathematics. The author gives an extensive survey of Coxeter groups and of buildings, drawing from the contents of many books. She gives frequent and detailed references to books on her list of references, making them a lively and essential part of her presentation. She emphasizes [\textit{M. W. Davis}, The geometry and topology of Coxeter groups. Princeton, NJ: Princeton University Press (2008; Zbl 1142.20020)] and [\textit{J. E. Humphreys}, Reflection groups and Coxeter groups. Cambridge etc.: Cambridge University Press (1990; Zbl 0725.20028)] for Coxeter groups and [\textit{K. S. Brown}, Buildings. New York etc.: Springer-Verlag (1989; Zbl 0715.20017)] and [Humphreys, loc. cit.] for buildings. The author sets the stage in Chapter 1 ``Examples and motivation''. Here, she presents examples of geometric reflection groups such as dihedral groups, triangle groups and isometry groups of classical polytopes accompanied by colourful enlightening figures. Triangle groups and isometry groups of classical polytopes are studied. Colourful figures, display barycentric subdivisions of the cube and the dodecahedron. She introduces symmetric groups, tilings of Euclidean space by cubes, tilings of the hyperbolic plane by right-angled polygons and tilings of hyperbolic 3-space by right-angled dodecahedra. There are colourful figures of tessellations, barycentric subdivision of the cube and the dodecahedron. Let \(I\) be a finite indexing set and let \(S = \{s_i\}_{i \in I}\). Let \(M = (m_{i j})_{i, j \in I}\) be a matrix such that \(m_{ii} = 1\) for all \(i \in I\); \(m_{ij} = m_{ji}\) for all \(i,j \in I\); and \(m_{ij} \in \{2, 3, 4, \ldots\} \cup \{\infty \}\) for all distinct \(…
reviewed by: Erich W. Ellers
Mathematics Subject Classification ID: 20-02
Mathematics Subject Classification ID: 20E42
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 57M07
zbMATH DE Number: 6891810
zbMATH Keywords: Coxeter group
zbMATH Keywords: reflection group
zbMATH Keywords: Tits representation
zbMATH Keywords: basic construction
zbMATH Keywords: simplicial complex
zbMATH Keywords: Davis complex
zbMATH Keywords: building
zbMATH Keywords: chamber system
zbMATH Keywords: \(BN\)-pair
MaRDI profile type: Publication
OpenAlex ID: W2802660567
description: scientific article; zbMATH DE number 6891810