Half-dimensional collapse of ends of manifolds of nonpositive curvature (Q2009023)

Let \(M\) be a complete non-compact Riemannian manifold of dimension \(n\) with finite volume. Assume that the sectional curvature satisfies \(-1< k \le0\) and that there is a positive lower bound to the length of any geodesically embedded circle. Then one has (Gromov-Schroeder) that \(M\) is tame, i.e., the thin part of \(M_{<\epsilon}\) has finitely many components and each component is topologically a product of a closed manifold of dimension \(n-1\) with a ray. In this setting, the authors construct a geometric analogue of the rational Tits building and show that this analogue has dimension less than \([\frac{n}{2}]\).