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

Abstract

The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component H odd (4) the only GL + (2, R) orbit closures are closed orbits, the Prym locusQ(3, −1 3 ), and H odd (4).Together with work of Matheus-Wright, this implies that there are only finitely many non-arithmetic closed orbits (Teichmüller curves) in H odd (4) outside of the Prym locus.

Rank two affine submanifolds in ℋ(2,2) and ℋ(3,1)

“…This conjecture was already known to be true in genus two due to the work of [McM07]. The first results in higher genus were due to [NW14] and [ANW13], and they affirmed this conjecture in genus three for the two connected components of the stratum H(4): H hyp (4) and H odd (4), respectively. We use these results as a starting point toward the goal of classifying all higher rank affine manifolds in genus three.…”

“…Affine Manifold Techniques: First, both [NW14] and [ANW13] heavily relied on the use of [SW04] to find a horizontally periodic translation surface in every orbit closure. However, this was not sufficient for our purposes in this paper because when we find a translation surface with a "partially periodic" foliation (in the sense that there are horizontal cylinders, but they do not fill the surface) and some desirable properties, then we would like to preserve these properties while moving to a translation surface that is horizontally periodic.…”

“…Proposition 2.16. This allows for a significant departure from the techniques of [NW14] and [ANW13], which did not at all rely on any classification result in lower genus. On the other hand, the present work is built upon the results in H(4) by degenerating to H(4) and drawing conclusions from this degeneration, cf.…”

The boundary of an affine invariant submanifold

“…It would be interesting to classify non-arithmetic rank 2 orbit closures in H(6), as we have implied might be possible above. We hope that recent efforts to classify higher rank affine invariant submanifolds in low genus might be extended using the new tools in this paper [NW14, ANW16,AN].…”

From rational billiards to dynamics on moduli spaces

“…It is at present a major open problem to classify GL(2, R) orbit closures. Progress is ongoing using techniques based on flat geometry and dynamics; see for example [Wri14,NW14,ANW]. One of the key tools is the author's Cylinder Deformation Theorem, which allows one to produce deformations of a translation surface that remain in the GL(2, R) orbit closure, without actually computing any surfaces in the GL(2, R) orbit [Wri15a].…”