On commensurability of quadratic differentials

TL;DR

This paper investigates the structure of commensurability classes of quadratic differentials on surfaces, establishing the existence of unique minimal elements and exploring their relation to fibered commensurability.

Contribution

It proves that each commensurability class contains a unique orbifold quadratic differential and relates this to fibered commensurability concepts.

Findings

Each commensurability class has a unique orbifold element.

The natural order by covering relation is well-defined within classes.

Connections between commensurability and fibered commensurability are established.

Abstract

We consider commensurability of quadratic differentials on surfaces. Each commensurability class has a natural order by the covering relation. We show that each commensurability class contains a unique (orbifold) element. We also discuss the relationship between commensurability of quadratic differentials and fibered commensurability, a notion introduced by Calegari-Sun-Wang.