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

In this paper, we investigate the closure of a large class of Teichmüller discs in the stratum Q(1, 1, 1, 1){\mathcal{Q}(1, 1, 1, 1)} or equivalently, in a GL⁺₂(\mathbbR){{\rm GL}^+_2(\mathbb{R})} -invariant locus L{\mathcal{L}} of translation surfaces of genus three. We describe a systematic way to prove that the GL⁺₂(\mathbbR){{\rm GL}^+_2(\mathbb{R})} -orbit closure of a translation surface in L{\mathcal{L}} is the whole locus L{\mathcal{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 Teichmüller discs obtained by the Thurston–Veech’s construction with a trace field of degree three which are 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{\mathcal{L}} , contrary to the situation in genus two. We also study completely periodic directions on translation surfaces in L{\mathcal{L}} . For instance, we prove that completely periodic directions are dense on surfaces obtained by the Thurston–Veech’s construction.