Completely periodic directions and orbit closures of many pseudo-Anosov Teichmueller discs in Q(1,1,1,1)

TL;DR

This paper analyzes the orbit closures of Teichmüller discs in a specific stratum, using periodic directions and Ratner's theorem, revealing new examples and density results in genus three translation surfaces.

Contribution

It introduces a systematic method to prove orbit closures are entire loci in genus three, applying Ratner's theorem to analyze periodic directions on translation surfaces.

Findings

Orbit closures can be the entire locus in genus three.

Constructs infinite series of non-Veech Teichmüller discs.

Shows density of completely periodic directions in certain surfaces.

Abstract

In this paper, we investigate the closure of a large class of Teichm\"uller discs in the stratum Q(1,1,1,1) or equivalently, in a GL^+_2(R)-invariant locus L of translation surfaces of genus three. We describe a systematic way to prove that the GL^+_2(R)-orbit closure of a translation surface in L is the whole of L. The strategy of the proof is an analysis of completely periodic directions on such a surface and an iterated application of Ratner's theorem to unipotent subgroups acting on an ``adequate'' splitting. This analysis applies for example to all Teichmueller discs stabilized obtained by Thurston's construction with a trace field of degree three which moreover ``obviously not Veech''. We produce an infinite series of such examples and show moreover that the favourable splitting situation does not arise everywhere on L, contrary to the situation in genus two. We also study completely periodic directions on translation surfaces in L. For instance, we prove that completely periodic directions are dense on surfaces obtained by Thurston's construction.