Geodesic
Veech surfaces and complete periodicity in genus two
Calta, Kariane1
1 Department 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 
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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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