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

Pascal Hubert, Erwan Lanneau, Martin Möller

Number | 32 |
---|---|

Author | Martin Möller |

Year | 2007 |

In this paper, we investigate the closure of a large class of Teichmüller
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.