On the asymptotic geometry of nonpositively curved manifolds (Q1380493)

Let \(M\) be a non-flat complete simply connected manifold with \(K_M\leq 0\). Suppose that \(M\) admits a cocompact discrete group of isometries. For \(p\in M\), denote by \(s(r,p)\) the volume of the geodesic sphere of radius \(r\) about \(p\). The main result of the paper is \[ s(r,p)\approx r^{{k- 1\over 2}}\cdot e^{hr}, \] where \(k\) is the geometric rank and \(h\) is the volume entropy of \(M\).