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

TL;DR

This paper uses exchange-graph methods to analyze the fundamental groups of genus-zero quadratic differential strata, extending combinatorics to weighted mixed-angulations and providing explicit presentations for cases with four singularities.

Contribution

It introduces a new combinatorial framework using exchange graphs and weighted mixed-angulations to study the topology of quadratic differential strata, especially for genus-zero cases.

Findings

Exchange graphs generate the fundamental group.

Relations extend from simple-zero to higher-order zero cases.

Explicit fundamental group presentations for genus-zero, four singularities.

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.