Translation Surfaces with No Convex Presentation

We give infinite lists of translations surfaces with no convex presentations. We classify the surfaces in the stratum H(2) which do not have convex presentations, as well as those with no strictly convex presentations. We show that in H(1, 1), all surfaces in the eigenform loci E4, E9 or E16 have no strictly convex presentation, and that the list of surfaces with no convex presentations in H(1, 1) r (E4 ∪ E9 ∪ E16) is finite and consists of square-tiled surfaces. We prove the existence of non-lattice surfaces without strictly convex presentations in all of the strata H(hyp)(g − 1, g − 1). A translation surface is a union of polygons with pairs of parallel edges identified by translation, up to cut and paste equivalence. These structures have been intensively studied in recent years in connection with questions in group theory, geometry, complex analysis, and dynamics — we recall the definitions in §1 and refer to the recent surveys [MT, Zo] for detailed definitions and discussions of work on translation surfaces. The cut and paste equivalence means that the same translation surface has many presentations as unions of polygons. It is well-known (see [Vi, §12]) that a connected translation surface may be presented as a single polygon with an even number of edges identified pairwise. A natural question is whether there is such a presentation, in which the polygon can be taken to be convex. In his 1992 Hayashibara Forum Lecture, Veech [V2] was the first to exhibit surfaces with no presentations as convex polygons. We review his examples and give new ones in §2. These include the Veech double n-gons, the Escher staircases, the Ward examples, and the Bouw-Möller examples. Translation surfaces are naturally grouped in strata, which are moduli spaces of translation surfaces for which the combinatorics of singularities is fixed. Although the constructions in §2 yield infinitely many examples, they only give rise to finitely many in each stratum. In this paper we take up the question of classifying surfaces with no convex presentations in the simplest strata, namely the genus two strata H(2) and H(1, 1). It will be useful to distinguish between convex and strictly convex polygons. Although, strictly speaking, polygons are never strictly convex, we will say that a polygon is strictly convex if none of its vertices is in the convex hull of the other vertices; equivalently, the internal angles at vertices of the polygon are strictly less than π. In making this definition we always assume that polygons have no spurious vertices, in other words, edges of polygons are not strictly contained in larger straight segments on which the identification map of edges extends continuously. We will see that strictly convex presentations only arise in the hyperelliptic components of the strata H(2g − 2) and H(g − 1, g − 1). Let G def = SL2(R). There is a natural G-action on each stratum of translation surfaces, and, as noted by Veech [V2], the property of having no (strictly) convex