Collapsing of quotient spaces of \(\text{SO} (n)\backslash \text{SL} (n,\mathbb{R})\) at infinity
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 \(|T|\) whose vertices are the elements of \(\mathcal S\). A congruence subgroup \(\Gamma\) acts on \(\mathcal V\) by conjugation, and this action extends naturally to an action of \(\Gamma\) on \(|T|\) whose quotient space is denoted by \(|T_\Gamma|\). The author's first main result is the construction of a family of (minimizing) geodesic rays in \(M = \text{SO} (n,\mathbb{R}) \setminus \text{SL}(n,\mathbb{R})/\Gamma\) that correspond bijectively to the points of \(|T_\Gamma|\). The author poses the question of whether all geodesic rays in \(M\) are asymptotically equivalent to one of the rays in this family. The author's second main result is that the limit as \(t \to \infty\) in the Gromov-Hausdorff topology of the pointed metric spaces \((M(1/t)g, q_0)\) equals \((C |T_\Gamma|, d)\), where \(g\) is the original Riemannian metric on \(M\) and \(q_0\) is the point of \(M\) determined by the identity coset \(I \cdot \text{SL}(n,\mathbb{R})\) in \(\text{SO} (n,\mathbb{R})\setminus \text{SL}(n,\mathbb{R})\).
scientific article; zbMATH DE number 844451