Geodesic
Veech surfaces and complete periodicity in genus two
Calta, Kariane1
1Department of Mathematics, Cornell University, Ithaca, New York 14853
Journal of the American Mathematical Society, Tome 17 (2004) no. 4, pp. 871-908

Voir la notice de l'article provenant de la source American Mathematical Society

We present several results pertaining to Veech surfaces and completely periodic translation surfaces in genus two. A translation surface is a pair $(M, \omega )$ where $M$ is a Riemann surface and $\omega$ is an Abelian differential on $M$. 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 $\omega$. A direction $v$ on a translation surface is completely periodic if any trajectory in the direction $v$ is either closed or ends in a singularity, i.e., if the surface decomposes as a union of cylinders in the direction $v$. 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 $SL(2, \mathbb {R})$ 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 $J$-invariant of Kenyon and Smillie to obtain a classification of all Veech surfaces in the space ${\mathcal H}(2)$ 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.
DOI : 10.1090/S0894-0347-04-00461-8

Calta, Kariane  1

1 Department of Mathematics, Cornell University, Ithaca, New York 14853 
Calta, Kariane. Veech surfaces and complete periodicity in genus two. Journal of the American Mathematical Society, Tome 17 (2004) no. 4, pp. 871-908. doi: 10.1090/S0894-0347-04-00461-8
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