Length bounds for hyperbolic Heegaard splittings

Sometimes the topology of a closed 3-manifold is dominated by a single closed surface together with some gluing data. A general challenge is to extract from such data information about the topology and the geometry of the manifold keeping in mind questions like: When does the manifold satisfy the hypothesis of the hyperbolization theorem of Thurston and Perelman? Is it possible to give a formula that turns the combinatorial data of the gluing into geometric invariants of the hyperbolic metric? In this talk, I will explain some recent joint work with Peter Feller and Alessandro Sisto addressing these questions for hyperbolic Heegaard splittings (a Heegaard splitting is a decomposition of a closed 3-manifold into two handlebodies intersecting along their common boundary, the so-called Heegaard surface). In particular, we are able to detect and control the length of some very short closed geodesics in the manifold (giving some novel information about the thick-thin decomposition of the manifold).