# Higher spin mapping class groups and strata of Abelian differentials over Teichm{\"u}ller space

## TL;DR

This paper classifies the connected components of strata of abelian differentials over Teichmüller space for genus g ≥ 5, revealing that non-hyperelliptic components are distinguished by r-spin structures, and computes related monodromy groups.

## Contribution

It provides a complete classification of these components using r-spin structures and explicitly determines the associated monodromy groups within the mapping class group.

## Key findings

- Classification of non-hyperelliptic components by r-spin structures
- Explicit generators for r-spin stabilizer subgroups
- Connections to flat and toric geometry

## Abstract

For $g \ge 5$, we give a complete classification of the connected components of strata of abelian differentials over Teichm\"uller space, establishing an analogue of Kontsevich and Zorich's classification of their components over moduli space. Building off of work of the first author (arXiv:1901.05482), we find that the non-hyperelliptic components are classified by an invariant known as an $r$-spin structure. This is accomplished by computing a certain monodromy group valued in the mapping class group. To do this, we determine explicit finite generating sets for all $r$-spin stabilizer subgroups of the mapping class group, completing a project begun by the second author (arXiv:1710.08042). Some corollaries in flat geometry and toric geometry are obtained from these results.