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In this paper, we introduce a particularly nice family of locally CAT()1) spaces, which we call hyperbolic P-manifolds. For X 3 a simple, thick hyperbolic P-manifold of dimension 3, we show that certain subsets of the boundary at infinity of the universal cover of X 3 are characterized topologically. Straightforward consequences include a version of Mostow rigidity, as well as quasi-isometry rigidity for these spaces.

Abstract. In this paper, we introduce the notion of an EZ-structure on a group, an equivariant version of the Z-structures introduced by Bestvina [4]. Examples of groups having an EZstructure include (1) torsion free δ-hyperbolic groups, and (2) torsion free CAT(0)-groups.Our first theorem shows that any group having an EZ-structure has an action by homeomorphisms on some (D n , ), where n is sufficiently large, and is a closed subset of ∂D n = S n−1 . The action has the property that it is proper and cocompact on D n − , and that if K ⊂ D n − is compact, that diam(gK) tends to zero as g → ∞. We call this property ( * ).Our second theorem uses techniques of to show that the Novikov conjecture holds for any torsion-free discrete group satisfying condition ( * ) (giving a new proof that torsion-free δ-hyperbolic and CAT(0) groups satisfy the Novikov conjecture).Our third theorem gives another application of our main result. We show how, in the case of a torsion-free δ-hyperbolic group , we can obtain a lower bound for the homotopy groups π n (P (B )), where P (·) is the stable topological pseudo-isotopy functor.

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We study compact Riemannian manifolds .M; g/ for which the light from any given point x 2 M can be shaded away from any other point y 2 M by finitely many point shades in M . Compact flat Riemannian manifolds are known to have this finite blocking property. We conjecture that amongst compact Riemannian manifolds this finite blocking property characterizes the flat metrics. Using entropy considerations, we verify this conjecture amongst metrics with nonpositive sectional curvatures. Using the same approach, K Burns and E Gutkin have independently obtained this result. Additionally, we show that compact quotients of Euclidean buildings have the finite blocking property.On the positive curvature side, we conjecture that compact Riemannian manifolds with the same blocking properties as compact rank one symmetric spaces are necessarily isometric to a compact rank one symmetric space. We include some results providing evidence for this conjecture.