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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


An infinite surface with the lattice property I: Veech groups and coding geodesics

Author: W. Patrick Hooper
Journal: Trans. Amer. Math. Soc. 366 (2014), 2625-2649
MSC (2010): Primary 37D40, 37D50, 37E99, 32G15
Published electronically: December 11, 2013
MathSciNet review: 3165649
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Abstract: We study the symmetries and geodesics of an infinite translation surface which arises as a limit of translation surfaces built from regular polygons, studied by Veech. We find the affine symmetry group of this infinite translation surface, and we show that this surface admits a deformation into other surfaces with topologically equivalent affine symmetries. The geodesics on these new surfaces are combinatorially the same as the geodesics on the original.

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  • [ANSS02] Jon Aaronson, Hitoshi Nakada, Omri Sarig, and Rita Solomyak, Invariant measures and asymptotics for some skew products, Israel J. Math. 128 (2002), 93–134. MR 1910377 (2003f:37012), 10.1007/BF02785420
  • [BV12] Joshua Bowman and Férran Valdez, Wild singularities of translations surfaces, to appear in Israel Journal of Mathematics, 2012.
  • [FU11] Krzysztof Fraczek and Corinna Ulcigrai, Non-ergodic z-periodic billiards and infinite translation surfaces, Inventiones, DOI 10.1007/S00222-013-0482-Z.
  • [Ghy87] Étienne Ghys, Groupes d’homéomorphismes du cercle et cohomologie bornée, The Lefschetz centennial conference, Part III (Mexico City, 1984) Contemp. Math., vol. 58, Amer. Math. Soc., Providence, RI, 1987, pp. 81–106 (French, with English summary). MR 893858 (88m:58024)
  • [HHW10] W. Patrick Hooper, Pascal Hubert, and Barak Weiss, Dynamics on the infinite staircase, Discrete and Continuous Dynamical Systems - Series A 33 (2010), no. 9, 4341-4347.
  • [Hoo08] W. Patrick Hooper, Dynamics on an infinite surface with the lattice property, unpublished,, 2008.
  • [Hoo10] W. Patrick Hooper, The invariant measures of some infinite interval exchange maps, preprint,, 2010.
  • [Hoo12] W. Patrick Hooper, An infinite surface with the lattice property II: Dynamics of pseudo-Anosovs, in preparation, 2012.
  • [HS10] Pascal Hubert and Gabriela Schmithüsen, Infinite translation surfaces with infinitely generated Veech groups, J. Mod. Dyn. 4 (2010), no. 4, 715–732. MR 2753950 (2012e:37075), 10.3934/jmd.2010.4.715
  • [HW12a] W. Patrick Hooper and Barak Weiss, Generalized staircases: recurrence and symmetry, Ann. Inst. Fourier 62 (2012), no. 4, 1581-1600 (English. French summary).
  • [HW12b] Pascal Hubert and Barak Weiss, Ergodicity for infinite periodic translation surfaces, 2012, to appear in Compositio Math.
  • [MT98] Katsuhiko Matsuzaki and Masahiko Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. Oxford Science Publications. MR 1638795 (99g:30055)
  • [RT12] David Ralston and Serge Troubetzkoy, Ergodic infinite group extensions of geodesic flows on translation surfaces, Journal of Modern Dynamics 6 (2012), no. 4, 477-497.
  • [RT13] -, Ergodicity of certain cocycles over certain interval exchanges, Discrete and Continuous Dynamical Systems 33 (2013), 2523-2529.
  • [Thu97] William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR 1435975 (97m:57016)
  • [Vee89] W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), no. 3, 553–583. MR 1005006 (91h:58083a), 10.1007/BF01388890
  • [Yoc10] Jean-Christophe Yoccoz, Interval exchange maps and translation surfaces, Homogeneous flows, moduli spaces and arithmetic, Clay Math. Proc., vol. 10, Amer. Math. Soc., Providence, RI, 2010, pp. 1–69. MR 2648692 (2012a:37004)
  • [Zor06] Anton Zorich, Flat surfaces, Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 2006, pp. 437–583. MR 2261104 (2007i:37070), 10.1007/978-3-540-31347-2_13

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Additional Information

W. Patrick Hooper
Affiliation: Department of Mathematics, City College of New York, New York, New York 10031

PII: S 0002-9947(2013)06139-9
Received by editor(s): November 3, 2010
Received by editor(s) in revised form: September 5, 2012
Published electronically: December 11, 2013
Additional Notes: This research was supported by N.S.F. Postdoctoral Fellowship DMS-0803013, N.S.F. Grant DMS-1101233, and a PSC-CUNY Award (funded by The Professional Staff Congress and The City University of New York).
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.