Boundaries of π1-injective surfaces

According to a result of A. Hatcher, just finitely many boundary slopes (isotopy classes of simple closed curves) can be realized as boundaries of incompressible, ∂-incompressible surfaces in a closed, compact, orientable, irreducible 3-manifold with boundary a single torus. We consider, in this paper, proper maps of surfaces (S, ∂S) into a 3-manifold (M, ∂M) which are injective on π1 and on relative π1, and which are embeddings on ∂S. We show that there exists a 3-manifold M, with boundary a single torus, in which every boundary slope is realized by the boundary of such a map. We prove a result interpreting the significance of boundary slopes of such surfaces for Dehn filling. More generally, we consider maps of surfaces S which are injective on π1 and on relative π1 as before, and which embed each component of ∂S, but do not necessarily embed all of ∂S. We show that there exists a 3-manifold with boundary a single torus admitting such a map of a connected surface simultaneously realizing an arbitrary finite set of boundary slopes. We also give examples generalizing the preceding constructions to the case where ∂M is a surface of higher genus.