Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fundamental Groups of Genus-$0$ Quadratic Differential Strata via Exchange Graphs

Jeonghoon So

Abstract

We investigate how exchange-graph techniques can be used to study the topology of strata of meromorphic quadratic differentials. The exchange graph provides natural generators for the fundamental group. By extending the combinatorics of triangulations to weighted mixed-angulations, we generalise the familiar relations arising in the simple-zero case and introduce an additional relation that appears only around higher-order zeroes. In the genus-zero case with four singularities, we show that these relations suffice to give explicit presentations of the fundamental group.

Fundamental Groups of Genus-$0$ Quadratic Differential Strata via Exchange Graphs

Abstract

We investigate how exchange-graph techniques can be used to study the topology of strata of meromorphic quadratic differentials. The exchange graph provides natural generators for the fundamental group. By extending the combinatorics of triangulations to weighted mixed-angulations, we generalise the familiar relations arising in the simple-zero case and introduce an additional relation that appears only around higher-order zeroes. In the genus-zero case with four singularities, we show that these relations suffice to give explicit presentations of the fundamental group.
Paper Structure (26 sections, 30 theorems, 188 equations, 32 figures)

This paper contains 26 sections, 30 theorems, 188 equations, 32 figures.

Table of Contents

  1. Introduction
  2. Braid Groups and Surfaces
  3. Braid and Surface Braid Groups
  4. Surfaces
  5. Mapping Class Groups
  6. Exchange Graphs
  7. Unoriented Exchange Graph of $\mathbf{S}$
  8. Oriented Exchange Graph of $\mathbf{S}$
  9. Exchange Graph of $\mathbf{S}_\Delta$
  10. Exchange Graph of $\mathbf{S}_\mathbf{w}$
  11. Braid Twist Group
  12. Quadratic differentials
  13. Framed Quadratic Differentials
  14. From Differential to Exchange Graph
  15. Wall and Chamber Structure
  16. ...and 11 more sections

Key Result

Proposition 1

There is an isomorphism of orbifold fundamental groups.

Figures (32)

  • Figure 1: A flip: replacing one diagonal of a quadrilateral by the other.
  • Figure 2: left: Square relation, right: Pentagon relation
  • Figure 3: left: Square relation, right: Pentagon relation
  • Figure 4: Hexagon relation
  • Figure 5: The forward flip in $\operatorname{EG}(\mathbf{S}_\Delta)$ and $\operatorname{EG}(\mathbf{S})$
  • ...and 27 more figures

Theorems & Definitions (84)

  • Proposition 1: Proposition \ref{['theorem:quadeg']}
  • Theorem 1: Theorem \ref{['thm:isomorphism-four-singularity']}
  • Definition 1: birman2016braids, Theorem 1.8
  • Theorem 2: lee2010positive, Thm. 1.1,Rem. 3.1
  • Definition 2: Fox-Neuwirth
  • Remark 1
  • Lemma 1
  • proof
  • Definition 3
  • Definition 4
  • ...and 74 more