Veech Groups of Loch Ness Monsters

Piotr Przytycki[1]; Gabriela Schmithüsen[2]; Ferrán Valdez[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)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 2, page 673-687
  • ISSN: 0373-0956

Abstract

top
We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of G L + ( 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.

How to cite

top

Przytycki, Piotr, Schmithüsen, Gabriela, and Valdez, Ferrán. "Veech Groups of Loch Ness Monsters." Annales de l’institut Fourier 61.2 (2011): 673-687. <http://eudml.org/doc/219716>.

@article{Przytycki2011,
abstract = {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.},
affiliation = {Polish Academy of Sciences Institute of Mathematics Śniadeckich 8 00-956 Warsaw (Poland); Karlsruhe Institute of Technology Institute of Algebra and Geometry 76128 Karlsruhe (Germany); U.N.A.M. Campus Morelia Morelia, Michoacán (Mexico)},
author = {Przytycki, Piotr, Schmithüsen, Gabriela, Valdez, Ferrán},
journal = {Annales de l’institut Fourier},
keywords = {Translation surfaces; infinite genus surfaces; Veech groups; translation surfaces},
language = {eng},
number = {2},
pages = {673-687},
publisher = {Association des Annales de l’institut Fourier},
title = {Veech Groups of Loch Ness Monsters},
url = {http://eudml.org/doc/219716},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Przytycki, Piotr
AU - Schmithüsen, Gabriela
AU - Valdez, Ferrán
TI - Veech Groups of Loch Ness Monsters
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 2
SP - 673
EP - 687
AB - 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.
LA - eng
KW - Translation surfaces; infinite genus surfaces; Veech groups; translation surfaces
UR - http://eudml.org/doc/219716
ER -

References

top
  1. Étienne Ghys, Topologie des feuilles génériques, Ann. of Math. (2) 141 (1995), 387-422 Zbl0843.57026MR1324140
  2. P. Hooper, Dynamics on an infinite surface with the lattice property, (2008) 
  3. P. Hoopert, P. Hubert, B Weiss, Dynamics on the infinite staircase surface, (2008) 
  4. P. Hubert, G. Schmithüsen, Infinite translation surfaces with infinitely generated Veech groups, (2008) Zbl1219.30019MR2753950
  5. Pascal Hubert, Howard Masur, Thomas Schmidt, Anton Zorich, Problems on billiards, flat surfaces and translation surfaces, Problems on mapping class groups and related topics 74 (2006), 233-243, Amer. Math. Soc., Providence, RI Zbl1307.37019MR2264543
  6. Pascal Hubert, Thomas A. Schmidt, An introduction to Veech surfaces, Handbook of dynamical systems. Vol. 1B (2006), 501-526, Elsevier B. V., Amsterdam Zbl1130.37367MR2186246
  7. John Smillie, Barak Weiss, Characterizations of lattice surfaces, Invent. Math. 180 (2010), 535-557 Zbl1195.57041MR2609249
  8. J. F. Valdez, Infinite genus surfaces and irrational polygonal billiards, Geom. Dedicata 143 (2009), 143-154 Zbl1190.37040MR2576299
  9. J. F. Valdez, Veech groups, irrational billiards and stable abelian differentials, (2009) Zbl1260.37024
  10. W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), 553-583 Zbl0676.32006MR1005006

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.