Geometry and algebra of hyperbolic 3-manifolds

There are two chapters. In the first, we prove that, given a compact orientable 3–manifold M whose boundary is a hyperbolic surface and a simple closed curve C in its boundary, every knot in M is homotopic to one whose complement admits a complete hyperbolic structure with totally geodesic boundary in which the geodesic representative of C is as small as you like. In the second, we construct a commensurably infinite collection of hyperbolic 3–manifolds that fiber over the circle and whose fundamental groups contain subgroups that are locally free and not free, answering a question of James Anderson.