  We prove that a Casson tower of height 4 contains a flat embedded disc
bounded by the attaching circle, and we prove disc embedding results for height
2 and 3 Casson towers which are embedded into a 4-manifold, with some
additional fundamental group assumptions. In the proofs we create a capped
grope from a Casson tower and use a refined height raising argument to
establish the existence of a symmetric grope which has two layers of caps, data
which is sufficient for a topological disc to exist, with the desired boundary.
As applications, we present new slice knots and links by giving direct
geometric constructions of slicing discs. In particular we construct a family
of slice knots which are potential counterexamples to the homotopy ribbon slice
conjecture.
