Foliations, flows, and dynamics in 3-manifolds

Abstract: Sutured manifold hierarchy is a powerful tool introduced by Gabai to study the topology of 3-manifolds. The length of a sutured manifold hierarchy gives us a measurement of how complicated the sutured manifold is. Also, using this tool, Gabai proved the existence of finite depth taut foliations. However, he didn’t discuss how finite the depth could be.

Sutured Instanton Floer homology was introduced by Kronheimer and Mrowka and is defined on balanced sutured manifolds. In this talk, I will explain how sutured Instanton Floer homology could offer us bounds on the minimal length of a sutured hierarchy and the minimal depth of a foliation for a fixed balanced sutured manifold.

Sutured manifolds provide a framework for studying taut foliations on a 3-manifold with a fixed homology class representing a closed leaf. In the context of sutured manifolds, tautness translates to the closed leaf being of minimal complexity, as measured by the Thurston norm. We explore a method for certifying tautness, by showing that the sutured manifold is homologically simple – a so-called 'rational homology product'. Most sutured manifolds do not have this form, but do always take the more general form of a 'twisted homology product', which incorporates a representation of the fundamental group. The question then becomes, how complicated of a representation is needed to realize a given sutured manifold as such?

We explore the case of sutured handlebodies, and see even among the simplest class of these, twisting is required. We give examples for which, when restricted to solvable representations, the twisting representation cannot be 'too simple'.

Abstract: We give a purely contact and symplectic geometric characterization of Anosov flows in dimension 3 and set up a framework to systematically use tools from contact and symplectic geometry and topology in the study of Anosov dynamics.

Abstract: Veering triangulations are a special class of ideal triangulations with a rather mysterious combinatorial definition. Their importance follows from a deep connection with pseudo-Anosov flows on 3-manifolds. Recently Landry, Minsky and Taylor introduced a polynomial invariant of veering triangulations called the taut polynomial. During the talk I will explain how and why it is connected to the Alexander polynomial of the underlying manifold.

Abstract: Despite the good comprehension that we have nowadays about the asymptotic dynamical behavior of a general Anosov flow, classification of the different orbital equivalent classes rest a major subject nowadays. In the particular case  of dimension three, a lot of different examples of Anosov flows can be constructed using surgery methods, which shows that the set of different equivalence classes is not at all simple to describe.

If we restrict to the special subfamily of transitive 3-dimensional Anosov flows, each flow has (many) associated open book decompositions of the 3-manifold with pseudo-Anosov monodromy, constructed via an immersed Birkhoff section. Since pseudo-Anosov homeomorphisms can be classified by terms of a combinatorial invariant, there is a hope of producing combinatorial invariants of the orbital equivalence class of these Anosov flows in terms of those available for pseudo-Anosov. The problem to solve is : Given two Birkhoff sections with pseudo-Anosov monodromy, how to determine if both correspond to the same flow?

We study a simpler question related to the previous one: Given an open book decomposition with pseudo-Anosov monodromy, can we determine whether or not the corresponding Anosov flow is a suspension Anosov flow ? In this talk we will explain how to translate this question into a problem of existence of some particular singular affine structures associated with the pseudo-Anosov monodromy. In turn, we can provide a natural bijection between the set of genus one Birkhoff sections of a suspension Anosov flow and these affine structures.

Abstract: According to H.Baik’s result, a fuchsian group G can be characterized by a collection of three G-invariant laminations  under some conditions. Recently, I focus on a tight pair that is a group with one invariant lamination and some extra conditions as a candidate which improves the above result. In this talk, I will briefly introduce some notions of laminations of the circle, and present some recent results about tight pairs. This work is joint with my advisor H.Baik. 

Abstract: I will discuss volumes of mapping tori associated to irreducible end periodic homeomorphisms of certain infinite-type surfaces, inspired by a theorem of Brock (in the finite-type setting) relating the volume of a mapping torus to the translation distance of its monodromy on the pants graph. This talk represents joint work with Elizabeth Field, Heejoung Kim, and Chris Leininger.

Abstract: Motivated by the L-space conjecture, I'll discuss a method for producing real line actions of 3-manifold groups from a certain class of taut foliations. The real line arises in a natural geometric way: as a quotient of the leaf space of the foliation. Along the way, I'll give an introduction to branching in the leaf space.