The object of this paper is to study GL 2 R orbit closures in hyperelliptic components of strata of abelian differentials. The main result is that all higher rank affine invariant submanifolds in hyperelliptic components are branched covering constructions, i.e. every translation surface in the affine invariant submanifold covers a translation surface in a lower genus hyperelliptic component of a stratum of abelian differentials. This result implies finiteness of algebraically primitive Teichmuller curves in all hyperelliptic components for genus greater than two. A classification of all GL 2 R orbit closures in hyperelliptic components of strata (up to computing connected components and up to finitely many nonarithmetic rank one orbit closures) is provided. Our main theorem resolves a pair of conjectures of Mirzakhani in the case of hyperelliptic components of moduli space. GL2R ORBIT CLOSURES IN HYPERELLIPTIC COMPONENTS 3Main Theorem 2. Let M be an orbit closure in H hyp (2g − 2) or H hyp (g − 1, g − 1) of complex dimension at least four. If dim M = 2r then M is a branched covering construction over H hyp (2r − 2) and if dim M = 2r + 1 then M is a branched covering construction over H hyp (r − 1, r − 1). The covers are branched over the zeros of the holomorphic one-forms and commute with the hyperelliptic involution.The letter r was chosen in the above theorem statement since it coincides with the rank of M.Remark 2. It is important to note that this result is about hyperelliptic components of strata and not about hyperelliptic loci of abelian differentials. An open problem related to this work is to find a classification of the orbit closures in other strata of the hyperelliptic locus, i.e. to analyze the GL 2 R dynamics of ΩH g (κ) for κ beyond (2g −2) and (g −1, g −1).As a corollary of Theorem 2 we have finiteness of algebraically primitive Teichmüller curves in hyperelliptic components. In the case of H hyp (g − 1, g − 1) this result was the main result of Möller [Möl08].Main Theorem 3. In H hyp (2g − 2) and H hyp (g − 1, g − 1) there are finitely many algebraically primitive Teichmüller curves for g > 2.Proof. Suppose not to a contradiction. Let C i be an infinite sequence of distinct algebraically primitive Teichmüller curves. By Eskin-Mirzakhani [EM] a subsequence equidistributes in a finite union of connected affine invariant submanifolds M = i M i . By Matheus-Wright [MW15] algebraically primitive Teichmüller curves cannot equidistribute in the connected component of any stratum when g > 2. Main Theorem 1 implies that no M i is higher rank since this would imply that C i is not geometrically primitive (and hence not algebraically primitive). Finally, no M i is rank one since these orbit closures only contain finitely many nonarithmetic Teichmüller curves by Lanneau-Nguyen-Wright [LNW]. Therefore, we have a contradiction.Work of Eskin, Filip, and Wright [EFW17] establishes the following.Theorem (Eskin, Filip, Wright [EFW17] Theorem 1.5). Any infinite collection of nonarithmetic rank one GL 2 R orbit clo...
We classify GL(2, R) invariant point markings over components of strata of Abelian differentials. Such point markings exist only when the component is hyperelliptic and arise from marking Weierstrass points or two points exchanged by the hyperelliptic involution. We show that these point markings can be used to determine the holomorphic sections of the universal curve restricted to orbifold covers of subvarieties of the moduli space of Riemann surfaces that contain a Teichmüller disk. The finite blocking problem is also solved for translation surfaces with dense GL(2, R) orbit.
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