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Full Title

Flow Equivalence Classes of Pseudo-Anosov Surface Homeomorphisms

Names

Billet, Robert (author)

Hironaka, Eriko (professor co-directing dissertation)

Petersen, Kathleen L. (professor co-directing dissertation)

Duke, Dennis (university representative)

Fenley, Sergio (committee member)

Heil, Wolfgang (committee member)

Florida State University (degree granting institution)

College of Arts and Sciences (degree granting college)

Department of Mathematics (degree granting department)

Date Issued

2016

Format

text

Abstract

This dissertation explores pseudo-Anosov elements of the mapping class group of an oriented surface from the point of view of fibered face theory. This theory runs dual to the classical way of thought where, rather than fixing a surface S and studying Mod(S), we study the set of all pseudo-Anosov mapping classes over all oriented surfaces. Let S be a connected, compact, oriented surface. The mapping class group of S, denoted Mod(S), is the group of orientation preserving homeomorphisms of S which act by the identity on θS considered up to isotopy. If no power of a mapping class leaves an essential curve invariant, the mapping class is pseudo-Anosov. In this case, the mapping class preserves an expanding and contracting foliation with expansion factor λ. The set of all pseudo-Anosov mapping classes admits a natural partition into flow equivalence classes. Such a class can be described as the surface cross sections transverse to a pseudo-Anosov flow in a hyperbolic fibered three manifold. Using the operation of Murasugi sum, we systematically study the flow equivalence classes that can be expressed as iterated Hopf plumbings on a disk in the 3-sphere. Such a surface is always a fiber surface for its boundary link. The data on how to attach the Hopf bands is conveniently packaged in a graph and, since the Coxeter element of this graph is, up to sign, the monodromy of the fiber surface, such links are called Coxeter links. The investigation splits into three main developments. The first result deals with the overall structure of the flow equivalence classes corresponding to Coxeter links as subspaces of the real vector space H¹(M;[the set of real numbers]), where M is the link exterior in the 3-sphere. The second result sheds light on a natural dynamically minimal representative in each class. We then give an algorithm that takes a class as input and outputs a multivariable polynomial which can be used to compute the expansion factor of any element contained in the class. By interpreting the mapping tori of the pseudo-Anosov mapping classes as link exteriors in the 3-sphere, we are able to identify the meridians of the link components with a basis for H¹(M;[integers]). With a few careful knot theoretic observations, we show that any surface with positive linking number to the original link is a fiber surface. With slightly stronger assumptions on the link, we show that the entire Thurston norm is determined by the norms of spanning surfaces for the individual components. It is easy to construct pseudo-Anosov mapping classes with small expansion factor on surfaces with high Euler characteristic. One way this can be achieved is by composing a periodic mapping class with a pseudo-Anosov map that is supported on a small subsurface. Since the flow equivalence class of a pseudo-Anosov homeomorphism contains maps supported on surfaces of arbitrarily high Euler characteristic, we consider the function λ❘x(S)❘. Using properties of this function and the above results, we find a natural minimizing element with respect to this function. The third result amounts to computing the Teichmüller polynomial for the fibered face in question. This can be a difficult process in general. Perhaps the most notable issues are explicitly computing the fixed cohomology and a train-track for a surface automorphism. After finding ways around these problems and others, we give the full algorithm to compute the Teichmüller polynomial.

Topics

Date of Defense

November 21, 2016.

Submitted Note

A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Bibliography Note

Includes bibliographical references.

Advisory Committee

Eriko Hironaka, Professor Co-Directing Dissertation; Kathleen Petersen, Professor Co-Directing Dissertation; Dennis Duke, University Representative; Sergio Fenley, Committee Member; Wolfgang Heil, Committee Member.

Publisher

Florida State University

Identifier

FSU_FA2016_Billet_fsu_0071E_13563

Use and Reproduction

This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). The copyright in theses and dissertations completed at Florida State University is held by the students who author them.

Rights Statement URI

Billet, R. (2016). Flow Equivalence Classes of Pseudo-Anosov Surface Homeomorphisms. Retrieved from http://purl.flvc.org/fsu/fd/FSU_FA2016_Billet_fsu_0071E_13563