Veech surfaces and complete periodicity in genus two

Abstract

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.

Euler characteristics of Teichmüller curves in genus two

“…Combining (2-18), (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19), and (2-20), we obtain…”

“….T 1;1 / is the stratum of stable Abelian differentials with one nonseparating polar node and one separating holomorphic node. (14)(15)(16)(17) There are four other strata consisting of stable Abelian differentials .X; !/ which vanish on some irreducible component of X . There is one such stratum corresponding to the curve system T 0;1 , one corresponding to the curve system T 2;1 , and two corresponding to the curve system T 1;1 .…”

“…Using the same symplectic basis f˛i;ˇig and 2 K D such that ˛1 D˛2 as above, Equation (7-9) again implies (7-14) and (7)(8)(9)(10)(11)(12)(13)(14)(15). Integrating Á over both sides yields (7-16) and (7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17), and substituting in (7-19) gives (7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) aÁ…”

“…Dividing by gcd.a; b; c/ and exponentiating yields (7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21).…”

“…Conversely if .Y; Á/ is in U minus the coordinate axes and is in the locus defined by (7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21), then (7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) holds modulo gcd.a; b; c/, and by changing q modulo gcd.a; b; c/, we can ensure that it holds exactly. It then follows that (7-16) and (7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) hold, and so .Y; Á/ is an eigenform.…”