The hyperbolic rank of homogeneous Hadamard manifolds (Q1849596)

Let \(H\) denote a homogeneous Hadamard manifold. For information on such manifolds we refer to [\textit{T. H. Wolter}, Geometry of homogeneous Hadamard manifolds, Int. J. Math. 2, 223-234 (1991; Zbl 0722.53046)]. Consider all geodesic hyperbolic spaces \(Y\) quasi-isometrically embedded into \(H\). In this paper, the author proves that \(\text{rank}_E(H)+\text{rank}_{\text{hyp}}(H)\geq\dim(H)\). It is important to mention that the above result was first proved by N. Brady and B. Farb, in 1998, for a Riemannian product space, followed by E. Leuzinger, in 2000, for symmetric spaces of non-compact type.