Archimedean toroidal maps and their edge covers
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Arnab Kundu
Abstract
The automorphism group of a map on a surface acts naturally on its flags (triples of incident vertices, edges, and faces). We will study the action of the automorphism group of a map on its edges. A map is semi-equivelar if all of its vertices have the same type of face-cycles. A semi-equivelar toroidal map refers to a semi-equivelar map embedded on a torus. If a map has k edge orbits under its automorphism group, it is referred to as a k-edge orbital or k-orbital. Specifically, it is referred to as an edge-transitive map if
Funding statement: This work is supported by the National Board for Higher Mathematics (under the Department of Atomic Energy, Government of India) Research Project Fund (No. 02011/9/2021-NBHM(R.P.)/R&D-II/9101). The fund aims to foster the development of higher mathematics in the country, formulate policies for mathematics development, assist in establishing and developing mathematical centers, and provide financial support for research projects, doctoral candidates, and postdoctoral scholars.
Acknowledgements
The authors are thankful to the referee for numerous useful comments and substantial improvement of the presentation of the article.
References
[1] U. Brehm and W. Kühnel, Equivelar maps on the torus, European J. Combin. 29 (2008), no. 8, 1843–1861. 10.1016/j.ejc.2008.01.010Search in Google Scholar
[2] D. M. Burton, Elementary Number Theory, 7th ed., Mc Graw Hill, New York, 2018. Search in Google Scholar
[3] H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Ergeb. Math. Grenzgeb. (3) 14, Springer, Berlin, 1980. 10.1007/978-3-662-21943-0Search in Google Scholar
[4] B. Datta and D. Maity, Semi-equivelar and vertex-transitive maps on the torus, Beitr. Algebra Geom. 58 (2017), no. 3, 617–634. 10.1007/s13366-017-0332-zSearch in Google Scholar
[5] B. Datta and D. Maity, Semi-equivelar maps on the torus and the Klein bottle are Archimedean, Discrete Math. 341 (2018), no. 12, 3296–3309. 10.1016/j.disc.2018.08.016Search in Google Scholar
[6] B. Datta and A. K. Upadhyay, Degree-regular triangulations of torus and Klein bottle, Proc. Indian Acad. Sci. Math. Sci. 115 (2005), no. 3, 279–307. 10.1007/BF02829658Search in Google Scholar
[7] K. Drach, Y. Haidamaka, M. Mixer and M. Skoryk, Archimedean toroidal maps and their minimal almost regular covers, Ars Math. Contemp. 17 (2019), no. 2, 493–514. 10.26493/1855-3974.1825.64cSearch in Google Scholar
[8] B. Grünbaum and G. C. Shephard, Tilings by regular polygons. Patterns in the plane from Kepler to the present, including recent results and unsolved problems, Math. Mag. 50 (1977), no. 5, 227–247. 10.1080/0025570X.1977.11976655Search in Google Scholar
[9] B. Grünbaum and G. C. Shephard, Edge-transitive planar graphs, J. Graph Theory 11 (1987), no. 2, 141–155. 10.1002/jgt.3190110204Search in Google Scholar
[10] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University, New York, 1979. Search in Google Scholar
[11] A. Hatcher, Algebraic Topology, Cambridge University, Cambridge, 2002. Search in Google Scholar
[12] I. Hubard, A. Orbanić, D. Pellicer and A. I. Weiss, Symmetries of equivelar 4-toroids, Discrete Comput. Geom. 48 (2012), no. 4, 1110–1136. 10.1007/s00454-012-9444-2Search in Google Scholar
[13] G. A. Jones, Edge-transitive embeddings of complete graphs, Art Discrete Appl. Math. 4 (2021), no. 3, Paper No. 3.03. 10.26493/2590-9770.1314.f6dSearch in Google Scholar
[14] J. Karabáš, 2020, https://www.savbb.sk/~karabas/edgetran.html. Search in Google Scholar
[15] J. Karabáš, R. Nedela and M. Skyvová, Census of oriented edge-transitive maps, (incomplete draft). Search in Google Scholar
[16] S. Katok, Fuchsian Groups, Chic. Lectures Math., University of Chicago, Chicago, 1992. Search in Google Scholar
[17] W. Kurth, Enumeration of Platonic maps on the torus, Discrete Math. 61 (1986), no. 1, 71–83. 10.1016/0012-365X(86)90029-4Search in Google Scholar
[18] A. Mednykh, Counting non-equivalent coverings and non-isomorphic maps for Riemann surfaces, 2005. Search in Google Scholar
[19] A. Orbanić, D. Pellicer, T. Pisanski and T. W. Tucker, Edge-transitive maps of low genus, Ars Math. Contemp. 4 (2011), no. 2, 385–402. 10.26493/1855-3974.249.3a6Search in Google Scholar
[20] J. Širáň, T. W. Tucker and M. E. Watkins, Realizing finite edge-transitive orientable maps, J. Graph Theory 37 (2001), no. 1, 1–34. 10.1002/jgt.1000.absSearch in Google Scholar
[21] S. Wilson, Edge-transitive maps and non-orientable surfaces, Math. Slovaca 47 (1997), 65–83. Search in Google Scholar
[22] https://en.wikipedia.org/wiki/Hermite_normal_form. Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Stability of solutions to obstacle problems with generalized Orlicz growth
- The Eisenstein cycles and Manin–Drinfeld properties
- Topological group actions by group automorphisms and Banach representations
- On abelian-by-cyclic Moufang loops
- Gromov ellipticity and subellipticity
- On the spectra of a class of Moran measures
- Finite-dimensional algebras not arising as blocks of group algebras
- A uniform Besov boundedness and the well-posedness of the generalized dissipative quasi-geostrophic equation in the critical Besov space
- Fourier integral operators with forbidden symbols on the Besov spaces
- Spectrality of Cantor–Moran measures with three-element digit sets
- Simple 𝔰𝔩 d+1-modules from Witt algebra modules
- Archimedean toroidal maps and their edge covers
- The smallest Mealy automaton generating an indicable regular branch group
- Global classical solution of the Cauchy problem to the 3D Benjamin–Bona–Mahony–Burgers-type equation with nonlocal control constraints
- Finite approximation properties of C*-modules III
