Single-cylinder square-tiled surfaces: Constructions and applications

Jeffreys, Luke (2020) Single-cylinder square-tiled surfaces: Constructions and applications. PhD thesis, University of Glasgow.

Abstract

This thesis investigates the combinatorial properties of square-tiled surfaces and studies the connections of these surfaces to the constructions of pseudo-Anosov homeomorphisms, and filling curves on punctured surfaces.

We begin by constructing, in every connected component of every stratum of the moduli space of Abelian differentials, square-tiled surfaces having a single vertical and single horizontal cylinder. We show that, for all but the hyperelliptic components, this can be achieved in the minimum number of squares required for a square-tiled surface in the ambient stratum. Moreover, for the hyperelliptic components, we show that the number of squares required is strictly greater and construct surfaces realising these bounds.
Using these surfaces, we demonstrate that pseudo-Anosov homeomorphisms optimising the ratio of Teichmüller to curve graph translation length are, in a reasonable sense, ubiquitous in the connected components of strata of Abelian differentials.

We consider the construction of filling pairs on punctured surfaces. We begin by determining the minimal intersection number of a filling pair on a genus two surface with an odd number, at least three, of punctures completing the work of Aougab-Huang and Aougab-Taylor. We then present a further application of the single-cylinder square-tiled surfaces constructed above by constructing filling pairs on punctured surfaces whose algebraic and geometric intersection numbers are equal.

Finally, we extend the constructions of single-cylinder square-tiled surfaces to certain strata of the moduli space of quadratic differentials.

In Chapter 1, we give the necessary background to describe the main results of this thesis. In Chapter 2, we prove the lemmas that are necessary for the construction of single-cylinder square-tiled surfaces in Chapter 3. Chapter 4 contains the construction of ratio-optimising pseudo-Anosov homeomorphisms, and the constructions of filling pairs on punctured surfaces are given in Chapter 5. In Chapter 6, we extend the constructions of Chapter 3 to certain strata of quadratic differentials. Finally, in Chapter 7, we present some remaining open questions and possible directions for future research. Appendix A gives an alternative proof of Proposition 3.2, and Appendix B contains the python code that realises the construction of Chapter 3.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Keywords: Abelian and quadratic differentials, square-tiled surfaces, pseudo-Anosov homeomorphisms, filling pairs.
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Supervisor's Name: Gadre, Dr. Vaibhav
• Date of Award: 2020
• Depositing User: Mr Luke Jeffreys
• Unique ID: glathesis:2020-81526
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 20 Jul 2020 07:37
• Last Modified: 08 Sep 2022 07:35
• Thesis DOI: 10.5525/gla.thesis.81526
• URI: https://theses.gla.ac.uk/id/eprint/81526