Higher spin mapping class groups and strata of Abelian differentials over Teichmüller space

(with Nick Salter)

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(last updated June 5, 2019)

For g ≥ 5, we give a complete classification of the connected components of strata of abelian differentials over Teichmüller space, establishing an analogue of a theorem of Kontsevich and Zorich in the setting of marked translation surfaces. Building off of work of the first author, 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 in a recent paper. Some corollaries in flat geometry and toric geometry are obtained from these results.