| When studying a 3-manifold, there are certain natural ways to characterize it. Examples include the manifold's geometric structure, the minimal genus of a Heegaard splitting, the structure of its fundamental group, and how it can be decomposed. While studying such questions, topologists have often turned to a large family of manifolds known as punctured-torus bundles. In this paper, we study the manifolds obtained by performing Dehn filling on hyperbolic punctured-torus bundles.;After reviewing their construction and basic properties, we use them to investigate three different areas. We first look at the connection between essential laminations and actions by the fundamental group, and show that there are infinitely many hyperbolic manifolds which admit no Reebless foliation but whose fundamental groups still act nontrivially on a simply-connected 1-manifold. This is done by combining existence of essential surfaces with work of John Baldwin.;We then turn to the relationship between Heegaard genus and rank of the fundamental group. In particular, we build on work of Kenneth Baker to calculate these invariants for all Dehn fillings of hyperbolic punctured-torus bundles which are along a slope intersecting each fiber once, as well as all fillings for a certain family of punctured-torus bundles.;Finally, we review the notion of cosmetic surgery, and characterize all exceptional cosmetic fillings on hyperbolic punctured-torus bundles, save for two pairs. We do so by using the computation of David Futer et al. of the geometric structure of the fillings, along with results of Darryl McCullough and the realization of particular fillings as double branched covers over 3-braids. |
