Corank and asymptotic filling-invariants for symmetric spaces

Abstract

Let X be a Riemannian symmetric space of noncompact type. We prove that there exists an embedded submanifold Y ⊂ X 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.

On the Filling Invariants at Infinity of Hadamard Manifolds

“…All the previously known results about the filling invariants, see [BF98,Ger94a,Leu00], as well as the new results in this paper suggest a connection between these invariants and the connectedness of the Tits boundary of a Hadamard manifold. Theorem 1.3 is one example.…”

“…That was proved by Leuzinger to be true for n = 3, see [Leu00]. We show that the same result does not hold anymore for n > 3.…”

“…Theorem A.1 (Leuzinger [Leu00]). If X is a rank k symmetric space of nonpositive curvature, then div k−1 (X) has exponential growth.…”

“…In section 7 we give the graph manifold example mentioned above. In appendix A we give a different and shorter proof of Leuzinger's Theorem, which was proved in [Leu00]. helpful discussions during the development of this paper, and also like to thank the University of Bonn, in particular Werner Ballmann for the opportunity to visit in the summer of 2003 when his interest began in the filling invariants.…”

The asymptotic rank of metric spaces

“…As a consequence of Theorems Related results have been obtained for symmetric spaces of non-compact type ( [6], [24], [16]) and for proper cocompact Hadamard spaces ( [29]) for the higher divergence invariants div k of Brady and Farb.…”