Geometry of quotient spaces of SO(3)\(\backslash\) SL(3,\(\mathbb{R}\)) by congruence subgroups
type: Item
instance of: scholarly article
title: Geometry of quotient spaces of SO(3)\(\backslash\) SL(3,\(\mathbb{R}\)) by congruence subgroups
zbMATH Open document ID: 0769.53033
DOI: 10.1007/BF01444727
author: Toshiaki Hattori
published in: Mathematische Annalen
publication date: 1993-03-03T00:00:00Z
full work available at URL: https://eudml.org/doc/164968
review text: Let \((M,g)\) be a complete, noncompact Riemannian manifold with finite volume and sectional curvature \(K \leq 0\). For each positive number \(t\) consider the complete metric \(g_ t = {1\over t}g\). Fix a point \(p\) of \(M\). It is an interesting problem to determine the Hausdorff limit of the pointed metric spaces (\((M,g_ t),p\)) as \(t\to \infty\) if such a limit exists. [For definitions and discussion see \textit{M. Gromov}, Structures métriques pour les variétés riemanniennes (Paris, 1981; Zbl 0509.53034).] Such a limiting metric space (\((X,d),q\)), if it exists, would describe the geometry of \(M\) near infinity in some sense. In certain cases the description of (\((X,d),q\)) is known and easy to state: 1) If \(M\) has sectional curvature bounded between two negative constants, then (\((X,d),q\)) is a cone with vertex \(q\) over \(k\) points, where \(k\) is the number of ends of \(M\). 2) Let \(M = M_ 1 \times M_ 2\), Riemannian product, where \(M_ 1\), \(M_ 2\) are as in 1) and each has only one end. Then (\((X,d),q\)) can be identified with the first quadrant in \(\mathbb{R}^ 2\). In this article the author considers the rank 2, locally symmetric space \(M = \Gamma_ m\setminus SL(3,\mathbb{R})/SO(3,\mathbb{R})\), where \(\Gamma_ m\) is the congruence subgroup of \(SL(3,\mathbb{Z})\) consisting of those elements \((g_{ij})\) of \(SL(3,\mathbb{Z})\) such that \(g_{ij} = \delta_{ij}\pmod m\). To describe (\((X,d),q\)) in this case we need some facts about \(M\). The space \(M\) is 5-dimensional and has a natural compactification \(\widetilde M\) due to Borel and Serre. The boundary \(\partial\widetilde{M}\) has a stratification that can be described by a graph \(| T_ \Gamma|\) such that the vertices of \(| T_ \Gamma|\) correspond to the 4-dimensional strata while the edges of \(| T_ \Gamma|\) correspond to the 3-dimensional strata of \(\partial \widetilde {M}\). The graph \(| T_ \Gamma|\) is a quotient space of the geometric realization of the Tits building…
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 126345
zbMATH Keywords: Hausdorff limit
zbMATH Keywords: pointed metric spaces
zbMATH Keywords: cone
zbMATH Keywords: congruence subgroup
zbMATH Keywords: Tits building
zbMATH Keywords: geodesic rays
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
cites work: Manifolds of nonpositive curvature
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OpenAlex ID: W1970348980
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description: scientific article; zbMATH DE number 126345