We present several results pertaining to Veech surfaces and completely periodic translation surfaces in genus two. A translation surface is a pair ( M , ω ) (M, \omega ) where M M is a Riemann surface and ω \omega is an Abelian differential on M M . Equivalently, a translation surface is a two-manifold which has transition functions which are translations and a finite number of conical singularities arising from the zeros of ω \omega . A direction v v on a translation surface is completely periodic if any trajectory in the direction v v is either closed or ends in a singularity, i.e., if the surface decomposes as a union of cylinders in the direction v v . Then, we say that a translation surface is completely periodic if any direction in which there is at least one cylinder of closed trajectories is completely periodic. There is an action of the group S L ( 2 , R ) SL(2, \mathbb {R}) on the space of translation surfaces. A surface which has a lattice stabilizer under this action is said to be Veech. Veech proved that any Veech surface is completely periodic, but the converse is false. In this paper, we use the J J -invariant of Kenyon and Smillie to obtain a classification of all Veech surfaces in the space H ( 2 ) {\mathcal H}(2) of genus two translation surfaces with corresponding Abelian differentials which have a single double zero. Furthermore, we obtain a classification of all completely periodic surfaces in genus two.

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We give explicit pseudo-Anosov homeomorphisms with vanishing Sah-Arnoux-Fathi invariant. Any translation surface whose Veech group is commensurable to any of a large class of triangle groups is shown to have an affine pseudo-Anosov homeomorphism of this type. We also apply a reduction to finite triangle groups and thereby show the existence of non-parabolic elements in the periodic field of certain translation surfaces.

We give continued fraction algorithms for each conjugacy class of triangle Fuchsian group of signature (3, n, ∞), with n ≥ 4. In particular, we give an explicit form of the group that is a subgroup of the Hilbert modular group of its trace field and provide an interval map that is piecewise linear fractional, given in terms of group elements. Using natural extensions, we find an ergodic invariant measure for the interval map. We also study Diophantine properties of approximation in terms of the continued fractions and show that these continued fractions are appropriate to obtain transcendence results.2010 Mathematics subject classification: primary 11J70; secondary 11K50, 11J17, 11J81, 20H10.

We study the action of the horocycle flow on the moduli space of abelian differentials in genus two. In particular, we exhibit a classification of a specific class of probability measures that are invariant and ergodic under the horocycle flow on the stratum H(1, 1).