From symmetric spaces to buildings, curve complexes and outer spaces.
type: Item
instance of: scholarly article
title: From symmetric spaces to buildings, curve complexes and outer spaces.
zbMATH Open document ID: 1264.20031
author: Lizhen Ji
published in: Innovations in Incidence Geometry
publication date: 2012-09-05T00:00:00Z
reviewed by: Oliver Goertsches
Mathematics Subject Classification ID: 20E42
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 14L35
zbMATH DE Number: 6077136
zbMATH Keywords: spherical Tits buildings
zbMATH Keywords: symmetric spaces
zbMATH Keywords: curve complexes
zbMATH Keywords: outer space
zbMATH Keywords: arithmetic groups
zbMATH Keywords: mapping class groups
zbMATH Keywords: outer automorphism groups of free groups
zbMATH Keywords: compactifications
MaRDI profile type: Publication
review text: Given a symmetric space \(X=G/K\) of noncompact type, with \(G\) the connected component of the isometry group of \(X\), the boundary of the geodesic compactification of \(X\) has the structure of a spherical Tits building, whose simplices are indexed by the proper parabolic subgroups of \(G\). The bulk of the paper under review is a survey of this well-known relation, various of its generalizations, and important applications. The unifying element is that in various settings, the structure at infinity of a noncompact space is related to a natural simplicial complex.NEWLINENEWLINE Among other things, the author discusses the Borel-Serre partial compactifications of noncompact symmetric spaces, as well as of the Teichmüller space \(T_g\) of a closed, orientable surface of genus \(\geq 2\), and of the outer space \(X_n\) of marked metric graphs with fundamental group equal to the free group of \(n\) generators. New results are presented in the last section of the paper: the author constructs a new simplicial complex, called the core graph complex, and discusses its relation to the geometry at infinity of \(X_n\).
description: scientific article; zbMATH DE number 6077136