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Zbl 0985.32008
Hubert, Pascal; Schmidt, Thomas A.
Invariants of translation surfaces. (English)
[J] Ann. Inst. Fourier 51, No.2, 461-495 (2001). ISSN 0373-0956; ISSN 1777-5310/e

A translation surface is a real 2-dimensional Euclidean manifold with cone singularities, equipped with an atlas for which the transition functions are translations. Such surfaces arise for instance in the study of billiards in polygons. W. Veech initiated the study of the diffeomorphisms of translation surfaces which are locally affine with respect to the translation structure. The differentials of these diffeomorphisms form a group which is now called the Veech group, and Veech proved that this group has discrete image in $\text{PSL}(2,\bbfR)$. He showed also that each of the Hecke triangle groups of odd index occur as Veech groups. \par In this paper, the authors show the following \par Theorem: The Hecke triangle groups of index 4 and 6 are not realizable as Veech groups. Furthermore, for each nonsquare natural number $N$, the PSL$(2,\bbfR)$ normalizer of the congruence group $\Gamma_0(N)$ is not realizable as a Veech group. \par There is a natural notion of a covering of a translation surface, which is called a balanced translation covering. Such a covering respects the translation structure and it respects also the singularities in the sense that the images and inverse images of singularities are singularities. In this paper, the authors consider trees of balanced coverings, and they show that there exist translation surfaces which have isomorphic Veech groups but which cannot lie in any common tree of balanced affine coverings. More precisely, they prove the following \par Theorem 2: The translation surface arising from the Euclidean triangle with angles $(\pi/2n,\pi/2n$, $(n-1)\pi/n)$ cannot share any common tree of balanced affine coverings with any surface which has a maximal Fuchsian group as Veech group. \par Theorem 3: The translation surfaces arising from the Euclidean triangles of angles $(\pi/18, \pi/18$, $8\pi/9)$ and $(2\pi/9, \pi/3, 4\pi/9)$ have isomorphic Veech groups but cannot share a common tree of balanced affine coverings.
[Athanase Papadopoulos (Strasbourg)]

MSC 2000:: *32G15 Teichmüller theory
57N16 Geometric structures on manifolds
57S25 Groups acting on specific manifolds
30F60 Teichmueller theory

Keywords: flat surfaces; billiards; Teichmüller disks; Hecke triangle groups; Veech groups; tree of balanced; affine coverings

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