Mirzakhani's covering conjecture for higher-rank arithmetic affine invariant submanifolds

An affine invariant submanifold is a submanifold of a stratum of abelian differentials cut out by real linear homogeneous equations in period coordinates. An affine invariant submanifold has higher rank when its rank is greater than one, and it is arithmetic in the sense used by the source. A branched covering construction is an orbit closure whose abelian differentials arise as pullbacks of quadratic differentials under holomorphic maps.

Mirzakhani's covering conjecture. Higher-rank arithmetic affine invariant submanifolds are either connected components of strata or branched covering constructions.

Together with Mirzakhani's arithmeticity conjecture, this claim is intended to explain the paucity of known geometrically primitive orbit closures. The paper proves the corresponding conclusions for the hyperelliptic components considered there, while the general conjecture remains open in the source.