no. 2
Veech Groups of Loch Ness Monsters
[Groupes de Veech du monstre du Loch Ness]
Przytycki, Piotr1 ; Schmithüsen, Gabriela2 ; Valdez, Ferrán3
1 Polish Academy of Sciences Institute of Mathematics Śniadeckich 8 00-956 Warsaw (Poland)
2 Karlsruhe Institute of Technology Institute of Algebra and Geometry 76128 Karlsruhe (Germany)
3 U.N.A.M. Campus Morelia Morelia, Michoacán (Mexico)
Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 673-687.

Nous classifions les groupes de Veech des surfaces de translation non compactes domestiquées. En particulier, nous prouvons que tous les sous groupes dénombrables de GL + (2,R) n’ayant pas d’éléments de norme plus petite que 1 apparaissent comme groupes de Veech des surfaces de translation non compactes domestiquées et dont le type topologique est celui du monstre du Loch Ness. Réciproquement, tout groupe de Veech d’une surface domestiquée est dénombrable ou bien conjugué à  un des trois groupes que nous précisons dans cet article.

We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of GL + (2,R) avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surface is either countable, or one of three specific types.

DOI : 10.5802/aif.2625
Classification : 20F65, 53A99
Keywords: Translation surfaces, infinite genus surfaces, Veech groups
Mots-clés : surfaces de translation, surfaces de genre infini, groupes de Veech

Przytycki, Piotr 1 ; Schmithüsen, Gabriela 2 ; Valdez, Ferrán 3

1 Polish Academy of Sciences Institute of Mathematics Śniadeckich 8 00-956 Warsaw (Poland)
2 Karlsruhe Institute of Technology Institute of Algebra and Geometry 76128 Karlsruhe (Germany)
3 U.N.A.M. Campus Morelia Morelia, Michoacán (Mexico) 
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Przytycki, Piotr; Schmithüsen, Gabriela; Valdez, Ferrán. Veech Groups of Loch Ness Monsters. Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 673-687. doi : 10.5802/aif.2625. https://aif.centre-mersenne.org/articles/10.5802/aif.2625/

[1] Ghys, Étienne Topologie des feuilles génériques, Ann. of Math. (2), Volume 141 (1995) no. 2, pp. 387-422 | DOI | MR | Zbl

[2] Hooper, P. Dynamics on an infinite surface with the lattice property (2008) (arXiv:0802.0189) | Zbl

[3] Hoopert, P.; Hubert, P.; Weiss, B Dynamics on the infinite staircase surface (2008) (http://www.math.bgu.ac.il//~barakw/papers/staircase.pdf)

[4] Hubert, P.; Schmithüsen, G. Infinite translation surfaces with infinitely generated Veech groups (2008) (preprint, http://www.cmi.univ-mrs.fr/~hubert/articles/hub-schmithuesen.pdf) | MR | Zbl

[5] Hubert, Pascal; Masur, Howard; Schmidt, Thomas; Zorich, Anton Problems on billiards, flat surfaces and translation surfaces, Problems on mapping class groups and related topics (Proc. Sympos. Pure Math.), Volume 74, Amer. Math. Soc., Providence, RI, 2006, pp. 233-243 | MR | Zbl

[6] Hubert, Pascal; Schmidt, Thomas A. An introduction to Veech surfaces, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 501-526 | MR | Zbl

[7] Smillie, John; Weiss, Barak Characterizations of lattice surfaces, Invent. Math., Volume 180 (2010) no. 3, pp. 535-557 | DOI | MR | Zbl

[8] Valdez, J. F. Infinite genus surfaces and irrational polygonal billiards, Geom. Dedicata, Volume 143 (2009) no. 1, pp. 143-154 | DOI | MR | Zbl

[9] Valdez, J. F. Veech groups, irrational billiards and stable abelian differentials (2009) (arXiv:0905.1591) | Zbl

[10] Veech, W. A. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., Volume 97 (1989) no. 3, pp. 553-583 | DOI | MR | Zbl

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