We i n vestigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of R k , Z k and Z k + . W e s h o w that they are either Haar measures or that every element of the action has zero metric entropy.

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Let M be a complete Riemannian manifold of bounded nonpositive sectional curvature and finite volume. We construct a topological Tits building A(~I) associated to the universal cover of M. If IV[ is irreducible and rank (M) >I 2, we show that A(~I) is a building canonically associated with a Lie group and hence that M is locally symmetric. I N T R O D U C T I O NLet M be a complete connected Riemannian manifold of bounded nonpositive sectional curvature and finite volume. For any geodesic y, let rank-( be the dimension of the space of parallel Jacobi fields along y. Let rank IV[ be the minimum of the ranks of all geodesics. This definition and the basic structure of such manifolds M with rank M >/ 2 were discussed in [BBE]

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