We prove a volume inequality for 3-manifolds having C 0 C^{0} metrics “bent” along a surface and satisfying certain curvature conditions. The result makes use of Perelman’s work on the Ricci flow and geometrization of closed 3-manifolds. Corollaries include a new proof of a conjecture of Bonahon about volumes of convex cores of Kleinian groups, improved volume estimates for certain Haken hyperbolic 3-manifolds, and a lower bound on the minimal volume of orientable hyperbolic 3-manifolds. An appendix compares estimates of volumes of hyperbolic 3-manifolds drilled along a closed embedded geodesic with experimental data.

Let d be a square free positive integer and O d the ring of integers in Q( √ −d). The main result of this paper is to show that the groups PSL(2, O d ) are subgroup separable on geometrically finite subgroups.

Abstract. We show that the problem of deciding whether a polygonal knot in a closed three-dimensional manifold bounds a surface of genus at most g is NP-complete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant C is NP-hard.

We show that for a hyperbolic knot complement, all but at most 12 Dehn fillings are irreducible with infinite word-hyperbolic fundamental group. AMS Classification numbers Primary: 57M50, 57M27Secondary: 57M25, 57S25