Hyperbolic structures from angled ideal triangulations
This talk will describe joint work with Ken Chan (University of Melbourne). Let $M$ be the interior of a compact orientable 3-manifold bounded by tori. Given an ideal triangulation $T$ of $M$, an `angle structure' on $T$ is a possibly singular hyperbolic structure on $M$ obtained by gluing together hyperbolic ideal tetrahedra so that the sum of dihedral angles around each edge is $2\pi$. The space of all such structures is a convex polytope with compact closure $A(T)$, and the total hyperbolic volume of the tetrahedra gives a continuous, concave function on $A(T)$. If this volume function is maximized at an interior point of $A(T)$, then the corresponding angled structure gives the (unique) complete hyperbolic structure on $M$. Hyperbolic structures for Dehn fillings on $M$ can be obtained by a similar volume maximization process.