On polyhedral retracts and compactifications of locally symmetric spaces
type: Item
instance of: scholarly article
title: On polyhedral retracts and compactifications of locally symmetric spaces
zbMATH Open document ID: 1052.22008
author: Enrico Leuzinger
published in: Differential Geometry and its Applications
publication date: 2004-08-16T00:00:00Z
review text: Let \(V:=\Gamma\setminus X\) be a locally symmetric space, where \(X\) is a Riemannian symmetric space of noncompact type and rank at least two and \(\Gamma\) a nonuniform, torsion-free, irreducible lattice in the group of isometries of \(X\). The results in this paper are based on the author's previous construction of the exhausting of \(V\) by Riemannian polyhedra \(V(s)\) (the Riemannian metric is the one induced from \(V\)), \(s\geq 0\), i.e., compact submanifolds with corners [Invent. Math. 121, 389--410 (1995; Zbl 0844.53040)]. The author proves that each \(V(s)\) is a strong deformation retract of \(V\) and \(V\) is homeomorphic to the interior of any \(V(s)\), which thus yields a compactification of \(V\). Equivalent such retractions were independently constructed by \textit{L. Saper} [Comment. Math. Helv. 72, 167--202 (1997; Zbl 0890.22003)]. This compactification approach is in contrast to other compactifications introduced by various authors [\textit{W. Baily} and \textit{A. Borel}, Ann. Math. (2) 84, 442--528 (1966; Zbl 0154.08602); \textit{A. Borel} and \textit{J.-P. Serre}, Comment. Math. Helv. 48, 436--491 (1973; Zbl 0274.22011); \textit{L. Saper}, loc. cit.; \textit{I. Satake}, Ann. Math. (2) 72, 555--580 (1960; Zbl 0146.04701); \textit{C. L. Siegel}, Zur Reduktionstheorie quadratischer Formen, Publ. Math. Soc. Japan 5 (1959; Zbl 0097.00901)]. Furthermore one shows that each retraction \(r_s:V\to V(s)\) has a \(\Gamma\)-invariant lift \(\hat{r}_s:X\to X(s)\), where \(X(s)\) is the universal covering space of \(V(s)\), and consequently the lattice \(\Gamma\) is isomorphic to \(\pi_1(V(s))\). The submanifold with corners \(X(s)\) of \(X\) is quasi-isometric to the discrete group \(\Gamma\) equipped with a left-invariant word metric. As an application the proof of a result announced by \textit{M. Gromov} [Geometric group theory (1993; Zbl 0841.20039)] asserting that the filling-volume functions of irreducible lattices in semisimple Lie groups of \(\math…
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 2091328
zbMATH Keywords: Riemannian symmetric space
zbMATH Keywords: locally symmetric space
zbMATH Keywords: discrete group
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: arithmetic group
zbMATH Keywords: reduction theory
zbMATH Keywords: Busemann function
zbMATH Keywords: polyhedral retract
zbMATH Keywords: Borel-Serre compactification
zbMATH Keywords: horoball
zbMATH Keywords: filling-volume function
zbMATH Keywords: Gromov-Hausdorff limit
reviewed by: Mircea Craioveanu
MaRDI profile type: Publication
full work available at URL: j.difgeo.2004.03.001
OpenAlex ID: W2025489915
Wikidata QID: Q115358070
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DOI: 10.1016/J.DIFGEO.2004.03.001
description: scientific article; zbMATH DE number 2091328