ABSTRACT. In this work we study the geodesic flow on n-dimensional ideal polyhedra and establish classical (for manifolds of negative curvature) results concerning the distribution of closed orbits of the flow.

In this paper we study the asymptotic behavior of cylindrical ends in compact foliated 3-manifolds and give a sufficient condition for these ends to spiral onto a toral leaf.

We prove that for any orientable connected surface S of finite type which is not a a sphere with at most four punctures or a torus with at most two punctures, the natural homomorphism from the extended mapping class group of S to the group of homeomorphisms of the space of geodesic laminations on S, equipped with the Thurston topology, is an isomorphism.

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Abstract. If X is a proper CAT (−1)-space and Γ a non-elementary discrete group of isometries acting properly discontinuously on X, it is shown that the geodesic flow on the quotient space Y = X/Γ is topologically mixing, provided that the generalized Busemann function has zeros on the boundary ∂X and the non-wandering set of the flow equals the whole quotient space of geodesics GY := GX/ Γ (the latter being redundant when Y is compact). Applications include the proof of topological mixing for (A) compact negatively curved polyhedra, (B) compact quotients of proper geodesically complete CAT (−1)-spaces by a one-ended group of isometries and (C) finite n-dimensional ideal polyhedra.