Abstract. In this article we introduce the notion of strongly poly-free groups and show that the Whitehead group of any such group vanishes. We also show that the pure braid groups are strongly poly-free. Now given any finitely presented torsion free group, to check the vanishing of the Whitehead group for this group, at first one would like to check if this group is the fundamental group of a complete nonpositively curved Riemannian manifold. We consider the case of pure braid group. It is not yet known if these groups can even act faithfully as a group of motions on a simply connected complete nonpositively curved Riemannian manifold. In this paper we introduce a new class of groups, namely the strongly poly-free groups which is a proper subclass of the class of poly-free groups and prove that the Whitehead group vanishes in this class of groups. We also show that the pure braid groups are strongly poly-free. We use some geometric results on 3-manifolds which fiber over the circle, the Fibered Isomorphism Theorem of Farrell and Jones and an induction argument to prove the result. Introduction.The paper is organized in the following way. In Section 1 we define strongly polyfree groups and state and prove that the Whitehead group of these groups are trivial. Section 2 is devoted to proving that the pure braid groups are strongly poly-free. We prove that the Whitehead group of the pure braid group of any compact (connected) surface except for the 2-sphere and the projective 2-plane vanishes in Section 3.ACKNOWLEDGMENT.

No abstract

Abstract. Let M be a rank 1 locally symmetric space of finite Riemannian volume. It is proved that the set of unit vectors on a non-constant C 1 curve in the unit tangent sphere at a point p e M for which the corresponding geodesic is bounded (relatively compact) in M, is a set of Hausdorff dimension 1.

We investigate the question of putting nonpositively curved metrics on doubles of finite volume real hyperbolic manifolds M , where we allow the boundary components to be identified via self-diffeomorphisms of ∂M .