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.