Geodesic rays in locally symmetric spaces
Let \(X\) be an irreducible Riemannian symmetric space of noncompact type and of rank \(\geq 2\). Consider a locally symmetric quotient \(V = \Gamma \backslash X\) by a discrete group of isometries \(\Gamma\) of \(X\). The arithmeticity theorem of Margulis asserts that if the volume of \(V\) is finite, then the lattice \(\Gamma\) is arithmetic. This imposes a \(\mathbb{Q}\)-structure on the group of isometries of \(X\), which is reflected in the geometry of \(V\). In the present paper we give a detailed description of geodesic rays in the ends (or cusps) of \(V\). Technically our approach is based on a differential geometric interpretation of well-known facts from the (algebraic) reduction theory of arithmetic groups. As an application we study the ``Tits geometry'' of such locally symmetric spaces \(V\).
ends of manifolds of nonpositive curvature
scientific article; zbMATH DE number 897466
scientific article