Geometry and Topology Seminar: "Three-manifolds, surfaces, digraphs, and entropy"
Abstract: I will describe two related projects from my first year at WashU concerning the Thurston norm on the second homology of a 3-manifold. This norm is a 3-manifold invariant with connections to many areas: geometric group theory, foliation theory, Floer theory, and more. There are some beautiful clues due to Thurston, Fried, Mosher, McMullen, and others that indicate there should be a dictionary between the combinatorics of the norm's polyhedral unit ball and the geometric/topological structures existing in the underlying manifold. However, this picture is very much incomplete and is mostly limited to the case when the manifold is a surface bundle over the circle.
I will explain some results which go beyond the surface bundle case to the more general setting of manifolds admitting veering triangulations, which are combinatorial objects I will define. First I will explain that one of these objects always cuts out the cone over a face of the norm ball in a natural way and computes the norm in this cone. Second, I will explain how one can use these objects to collate all surfaces representing classes in these cones which are compact leaves of taut foliations. Third, I will describe joint work with Yair Minsky and Samuel Taylor in which we use a veering triangulation to define a polynomial invariant, the "veering polynomial," generalizing the Teichmuller polynomial of McMullen and answering a question asked by Calegari and McMullen. This invariant can be obtained by computing the Perron polynomial of a certain directed graph.
Host: Yanli Song