Translation covers of some triply periodic Platonic surfaces
Authors:
Jayadev S. Athreya and Dami Lee
Journal:
Conform. Geom. Dyn. 25 (2021), 34-50
MSC (2020):
Primary 32G15, 57K20
DOI:
https://doi.org/10.1090/ecgd/357
Published electronically:
April 2, 2021
MathSciNet review:
4238630
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We study translation covers of several triply periodic polyhedral surfaces that are intrinsically Platonic. We describe their affine symmetry groups and compute the quadratic asymptotics for counting saddle connections and cylinders, including the count of cylinders weighted by area. The mathematical study of triply periodic surfaces was initiated by Novikov, motivated by the study of electron transport. The surfaces we consider are of particular interest as they admit several different explicit geometric and algebraic descriptions, as described, for example, in the second author’s thesis.
- J. Athreya, D. Aulicino, and P. Hooper, Platonic solids and high genus covers of lattice surfaces, Experimental Mathematics, (2020), pp.1–31.
- Jayadev S. Athreya, Jon Chaika, and Samuel Lelièvre, The gap distribution of slopes on the golden L, Recent trends in ergodic theory and dynamical systems, Contemp. Math., vol. 631, Amer. Math. Soc., Providence, RI, 2015, pp. 47–62. MR 3330337, DOI https://doi.org/10.1090/conm/631/12595
- H. S. M. Coxeter, Regular Skew Polyhedra in Three and Four Dimension, and their Topological Analogues, Proc. London Math. Soc. (2) 43 (1937), no. 1, 33–62. MR 1575418, DOI https://doi.org/10.1112/plms/s2-43.1.33
- V. Delecroix, C. Fougeron, and S. Lelievre, Surface Dynamics - SageMath package, Version 0.4.1, https://pypi.org/project/surface-dynamics/, 2019.
- Eugene Gutkin and Chris Judge, Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J. 103 (2000), no. 2, 191–213. MR 1760625, DOI https://doi.org/10.1215/S0012-7094-00-10321-3
- D. Lee, Geometric realizations of cyclically branched coverings over punctured spheres, Indiana University, (2018), Ph.D thesis.
- Sage website, https://www.sagemath.org.
- 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, DOI https://doi.org/10.1007/BF01388890
- William A. Veech, Siegel measures, Ann. of Math. (2) 148 (1998), no. 3, 895–944. MR 1670061, DOI https://doi.org/10.2307/121033
- Anton Zorich, How do the leaves of a closed $1$-form wind around a surface?, Pseudoperiodic topology, Amer. Math. Soc. Transl. Ser. 2, vol. 197, Amer. Math. Soc., Providence, RI, 1999, pp. 135–178. MR 1733872, DOI https://doi.org/10.1090/trans2/197/05
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Additional Information
Jayadev S. Athreya
Affiliation:
Department of Mathematics, University of Washington, Seattle, Washington 98195
MR Author ID:
663249
ORCID:
0000-0002-9317-6229
Email:
jathreya@uw.edu
Dami Lee
Affiliation:
Department of Mathematics, University of Washington, Seattle, Washington 98195
MR Author ID:
1229078
ORCID:
0000-0001-5924-4171
Email:
damilee@uw.edu
Received by editor(s):
December 18, 2019
Received by editor(s) in revised form:
September 22, 2020, and October 22, 2020
Published electronically:
April 2, 2021
Additional Notes:
The second author was supported by NSF Grant No. DMS-1440140
Article copyright:
© Copyright 2021
American Mathematical Society
