A highly useful technique for studying a 3-manifold is to decompose it into simpler pieces, such as tetrahedra, and to examine normal surfaces within the pieces. If the pieces admit additional data, e.g. an angle structure, then there are concrete geometric consequences for the manifold and the surfaces it contains. For example, we may determine conditions that guarantee the manifold is hyperbolic, estimate its volume, and identify quasifuchsian surfaces embedded within it.
In the first talk, I will briefly describe some history of these decompositions, including work of Thurston, Menasco, and Lackenby, and then describe how to generalise their work to extend results to broader families of 3-manifolds. For example, we may allow pieces that are not simply connected, glued along faces that are not disks. We give examples of manifolds with these structures, particularly families of knot and link complements.
In the second talk, using a generalisation of normal surfaces, angle structures, and combinatorial area, I will describe geometric consequences of these decompositions and further applications.
This is joint work with Josh Howie.