Parallelogram decompositions and generic surfaces in H^hyp(4)

Abstract

The space H hyp .4/ is the moduli space of pairs .M; !/, where M is a hyperelliptic Riemann surface of genus 3 and ! is a holomorphic 1-form having only one zero. In this paper, we first show that every surface in H hyp .4/ admits a decomposition into parallelograms and simple cylinders following a unique model. We then show that if this decomposition satisfies some irrational condition, then the GL C .2; R/-orbit of the surface is dense in H hyp .4/; such surfaces are called generic. Using this criterion, we prove that there are generic surfaces in H hyp .4/ with coordinates in any quadratic field, and there are Thurston-Veech surfaces with trace field of degree three over Q which are generic. 51H25; 37B05

The field of definition of affine invariant submanifolds of the moduli space of abelian differentials

“…by Hubert-Lanneau-Möller [HLM09a, HLM12, HLM09b] and also in H hyp (4) by Nguyen [Ngu11]. These results in L and H hyp (4) are complementary to ours; they do not provide a full measure set, but do provide many especially interesting and important examples not covered by our results.…”

The boundary of an affine invariant submanifold

“…Our methods, in particular Theorem 6.9, may clarify the extent to which the boundary of an affine invariant submanifold determines the affine invariant submanifold. HLM09,Ngu11].…”

“…See also [Cal04] for a different presentation of affine invariant submanifolds in genus 2. For some extensions of McMullen's techniques beyond genus 2, see [HLM12, HLM09,Ngu11].…”

Classification of higher rank orbit closures in ${\mathcal H^{\mathrm{odd}}(4)}$

“…Some generalizations of McMullen's techniques were made in genus 3 [HLM09, HLM12, Ngu11], showing that some especially interesting translation surfaces have dense orbits. This made it reasonable to conjecture that few truly new orbit closures exist in genus greater than 2.…”