[Paper Review] Rank One Orbit Closures in H^{hyp}(g-1,g-1)

Paul Apisa|arXiv (Cornell University)|Oct 16, 2017

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper classifies GL(2,R) orbit closures in hyperelliptic strata of abelian differentials in genus g > 2, proving that every orbit is either closed, dense, or contained in a locus of branched covers. Using cylinder deformation techniques and the classification of rank-one orbit closures in genus two, the authors show that all such orbit closures in ℋ^{hyp}(g−1,g−1) are either branched covers of eigenform loci in ℋ(1,1) or arise from arithmetic Teichmüller curves, resolving a key case in the dynamics of translation surfaces.

ABSTRACT

Every GL(2,R)-orbit in hyperelliptic components of strata of abelian differentials in genus greater than two is either closed, dense, or contained in a locus of branched covers.

Motivation & Objective

• To classify GL(2,R) orbit closures in hyperelliptic components of strata of abelian differentials in genus g > 2.
• To determine whether such orbit closures are closed, dense, or contained in loci of branched covers.
• To extend previous results on orbit closures of dimension >3 to the full classification of rank-one orbit closures.
• To establish that all non-arithmetic, rank-one orbit closures in ℋ^{hyp}(g−1,g−1) arise as branched covers of eigenform loci in ℋ(1,1).
• To complete the classification of invariant submanifolds in hyperelliptic strata by reducing the problem to known results in genus two.

Proposed method

• Utilizes cylinder deformation theory developed by Wright, particularly the twist space and cylinder preserving space constructions.
• Applies the rel deformation structure in hyperelliptic surfaces, where the rel is determined by alternating signs based on cylinder adjacency in the cylinder graph.
• Employs a standard position argument to normalize saddle connections and cylinders, enabling the construction of local isometries between surfaces.
• Reduces the classification problem to genus two via the use of horizontal and vertical cylinder tilings and their sub-equivalence classes.
• Uses the fact that orbit closures in genus two are fully classified by McMullen, particularly eigenform loci in ℋ(1,1), to deduce the structure of higher-genus orbit closures.
• Applies the Eskin-Mirzakhani-Mohammadi theorem on orbit closure linearity and the EFW finiteness result to conclude that only finitely many closed orbits exist outside branched cover loci.

Experimental results

Research questions

• RQ1 Are all GL(2,R) orbit closures in hyperelliptic strata of ℋ^{hyp}(g−1,g−1) for g > 2 either closed, dense, or contained in a locus of branched covers?
• RQ2 Can rank-one, non-arithmetic orbit closures in ℋ^{hyp}(g−1,g−1) be classified via reduction to genus two dynamics?
• RQ3 What is the role of the rel deformation in determining the structure of orbit closures in hyperelliptic components?
• RQ4 Do all such orbit closures arise as branched covers of eigenform loci in ℋ(1,1)?
• RQ5 How do cylinder deformation and sub-equivalence relations constrain the geometry of orbit closures in hyperelliptic strata?

Key findings

• All GL(2,R) orbits in hyperelliptic components of ℋ^{hyp}(g−1,g−1) for g > 2 are either closed, dense, or contained in a locus of branched covers.
• Every non-arithmetic, rank-one orbit closure in ℋ^{hyp}(g−1,g−1) is a branched covering construction of an eigenform locus in ℋ(1,1).
• The classification of orbit closures in genus two by McMullen implies that all such orbit closures in higher genus are either branched covers or arise from arithmetic Teichmüller curves.
• The rel deformation in these surfaces is uniquely determined by alternating signs based on cylinder adjacency, and this deformation generates the full twist space in standard position.
• The existence of a surface with a twist space containing the rel deformation allows for perturbation to ensure irrational modulus ratios, which is essential for ruling out spurious sub-equivalence relations.
• The orbit closure is a locus of branched covers of surfaces in ℋ(1,1), and since only non-arithmetic eigenforms exist in ℋ(1,1) of complex dimension 3, the orbit closure must be a non-arithmetic branched cover of such a locus.