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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[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**, 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**- [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*Dynamics on an infinite surface with the lattice property*, unpublished,`http://arxiv.org/abs/0802.0189`, 2008. - [Hoo10]
W. Patrick Hooper,
*The invariant measures of some infinite interval exchange maps*, preprint,`http://arxiv.org/abs/1005.1902`, 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**, 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**- [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****[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**, 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****[Zor06]**Anton Zorich,*Flat surfaces*, Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 2006, pp. 437–583. MR**2261104**, 10.1007/978-3-540-31347-2_13

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
37D40,
37D50,
37E99,
32G15

Retrieve articles in all journals with MSC (2010): 37D40, 37D50, 37E99, 32G15

Additional Information

**W. Patrick Hooper**

Affiliation:
Department of Mathematics, City College of New York, New York, New York 10031

Email:
whooper@ccny.cuny.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-2013-06139-9

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.