Geometry of quotient spaces of SO(3)\(\backslash\) SL(3,\(\mathbb{R}\)) by congruence subgroups
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 of \(SL(3,\mathbb{Q})\) by \(\Gamma\). We omit a more precise description. The author shows that \(| T_ \Gamma|\) can be identified bijectively with the elements of a certain family of asymptotic equivalence classes of geodesic rays in \(M\). There is evidence to suggest that this family may equal the entire collection of asymptotic equivalence classes of geodesic rays in \(M\). From the graph \(| T_ \Gamma|\) one can define a cone \(C| T_ \Gamma|\) with a distinguished vertex \(O\) by pasting together suitable Euclidean sectors. On \(C| T_ \Gamma|\) one defines a polyhedral metric \(d_ \Gamma\) which equals the Euclidean metric on each sector. One may now state the main result of the article: Theorem. Let \(M = \Gamma_ m\setminus SL(3,\mathbb{R})/SO(3,\mathbb{R})\) with the standard locally symmetric metric \(g\) with sectional curvature \(K \leq 0\). For any point \(p\) of \(M\) we have \(\lim_{t \to \infty}((M,g_ t),p) = ((C| T_ \Gamma|,d),O)\), where convergence takes place in the Hausdorff sense for pointed noncompact metric spaces.
scientific article; zbMATH DE number 126345
scientific article