Corank and asymptotic filling-invariants for symmetric spaces

Let X be a Riemannian symmetric space of noncompact type. We prove that there exists an embedded submanifold which is quasi-isometric to a manifold with strictly negative sectional curvature, which intersects a given flat F in a geodesic line and which satisfies dim(Y) — 1 = dim(X) — rank(X). This yields an estimate of the hyperbolic corank of X. As another application we show that certain asymptotic filling invariants of X are exponential. Submitted: February 1999, Revised version: November 1999.