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

Abstract

For g≥5, we give a complete classification of the connected components of strata of abelian differentials over Teichmüller space, establishing an analogue of Kontsevich and Zorich's classification of their components over moduli space. Building on work of the first author [2], 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 [18]. Some corollaries in flat geometry and toric geometry are obtained from these results.