Collapsing of quotient spaces of \(\text{SO} (n)\backslash \text{SL} (n,\mathbb{R})\) at infinity
type: Item
instance of: scholarly article
title: Collapsing of quotient spaces of \(\text{SO} (n)\backslash \text{SL} (n,\mathbb{R})\) at infinity
zbMATH Open document ID: 0844.53041
author: Toshiaki Hattori
published in: Journal of the Mathematical Society of Japan
publication date: 1996-03-11T00:00:00Z
review text: Let \(\Gamma\) denote a principal congruence subgroup of \(\text{SL}(n,\mathbb{Z})\) of level \(m\); that is, \(\Gamma = \{g = (g_{ij}) \in \text{SL} (n,\mathbb{Z}) : g_{ij} = \delta_{ij} \pmod m\}\). In this article the author investigates the geometry at infinity of the quotient space \(M = \text{SO}(n, \mathbb{R}) \setminus \text{SL}(n,\mathbb{R})/\Gamma\), which is a locally symmetric Riemannian manifold with sectional curvature \(K \leq 0\), rank \(n - 1\) and finite volume. The geometry of such spaces at infinity has been described algebraically by many authors. In this paper the language and arguments are also algebraic, but the results are formulated in terms of the space \(|T_\Gamma|\) and its associated metric space \((C |T_\Gamma|, d)\), where \(C|T_\Gamma|\) is a polyhedron constructed from the quotient \(|T_\Gamma|\) of the Tits building \(|T|\) by the action of the group \(\Gamma\), and \(d\) is a metric that is Euclidean on the maximal faces of \(C|T_\Gamma|\), each of which may be identified isometrically with a closed Weyl chamber in the space \(A = \mathbb{R}^{n - 1}\) of real diagonal matrices with trace zero. Here metric spaces are compared by means of the Gromov-Hausdorff metric and topology in which a sequence of pointed metric spaces \((M_n, p_n)\) converges to a pointed metric space \((M_0, p_0)\) if the metric balls in \(M_n\) of radius \(R\) and center \(p_n\) converge in an appropriate sense to the metric ball in \(M_0\) of radius \(R\) and center \(p_0\) for all \(R > 0\). The Tits building of \(G = \text{SL} (n,\mathbb{R})\) is a simplicial complex \(|T|\) whose vertices \(\mathcal V\) consist of the maximal parabolic \(\mathbb{Q}\)-subgroups of \(G\). A subset \(\mathcal S\) of \(\mathcal V\) is admissible if the intersection of all maximal parabolic \(\mathbb{Q}\)-subgroups in \(\mathcal S\) is also a parabolic \(\mathbb{Q}\)-subgroup of \(G\). Each admissible subset \(\mathcal S\) of \(\mathcal V\) determines a simplex in \(|T|\) whos…
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 20E99
zbMATH DE Number: 844451
zbMATH Keywords: parabolic subgroup
zbMATH Keywords: finite volume locally symmetric space
zbMATH Keywords: geometry at infinity
zbMATH Keywords: principal congruence subgroup
zbMATH Keywords: geodesic rays
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04720193
OpenAlex ID: W2078182377
DOI: 10.2969/JMSJ/04720193
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description: scientific article; zbMATH DE number 844451