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Quasi-flats and rigidity in higher rank symmetric spaces

In this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces. Our results are powerful enough to apply to noncocompact lattices in higher rank symmetric spaces, such as SL(n, 2), n > 3: Theorem 8.1 is a major step towards the proof of quasiisometric rigidity of such lattices ([E]). We also give a different, and effective, proof of the theorem of Kleiner-Leeb on the quasi-isometric rigidity of higher rank symmetric spaces ([KL]).