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Geometry and Symmetry Conference

GSC 2015: Discrete Geometry and Symmetry pp 147–170Cite as

Hexagonal Extensions of Toroidal Maps and Hypermaps

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 234))

Abstract

The rank 3 concept of a hypermap has recently been generalized to a higher rank structure in which hypermaps can be seen as “hyperfaces” but very few examples can be found in literature. We study finite rank 4 structures obtained by hexagonal extensions of toroidal hypermaps. Many new examples are produced that are regular or chiral, even when the extensions are polytopal. We also construct a new infinite family of finite nonlinear hexagonal extensions of the tetrahedron.

Dedicated to Egon Schulte on the occasion of his 60th birthday

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References

  1. F. Buekenhout, A.M. Cohen, Diagram Geometry. Related to classical groups and buildings. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 57 (Springer, Heidelberg, 2013), xiv+592 pp

    Google Scholar 

  2. J. Charles, Colbourn and Asia Ivić Weiss. A census of regular 3-polystroma arising from honeycombs. Discrete Math. 50, 29–36 (1984)

    Article  MathSciNet  Google Scholar 

  3. M. Conder, P. Dobcsányi, Determination of all regular maps of small genus. J. Combin. Theory Ser. B 81, 224–242 (2001)

    Article  MathSciNet  Google Scholar 

  4. M. Conder, I. Hubard, T. Pisanski, Constructions for chiral polytopes. J. Lond. Math. Soc. 2(77), 115–129 (2008)

    Article  MathSciNet  Google Scholar 

  5. M. Conder’s, website. http://www.math.auckland.ac.nz/~conder/. Last accessed 21 Sept 2014

  6. J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, Atlas of Finite Groups (Oxford U.P., Oxford, 1985)

    MATH  Google Scholar 

  7. H.S.M. Coxeter, Configurations and maps. Rep. Math. Colloq. 8, 18–38 (1948)

    MathSciNet  MATH  Google Scholar 

  8. H.S.M. Coxeter, Twisted Honeycombs, in Regional Conf. Ser. In Math. (American Mathematical Society, Providence, RI, 1970)

    Google Scholar 

  9. H.S.M. Coxeter, W.O.J. Moser, Generators and Relations for Discrete Groups, volume 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 4th edn. (Springer, Berlin, 1980)

    Google Scholar 

  10. H.S.M. Coxeter, G.C. Shephard, Regular \(3\)-complexes with toroidal cells. J. Comb. Theory Ser. B 22(2), 131–138 (1977)

    Article  MathSciNet  Google Scholar 

  11. L. Danzer, E. Schulte, Regulare Inzidenzkomplexe I. Geom. Dedicata 13, 295–308 (1982)

    Article  MathSciNet  Google Scholar 

  12. M. Dehon, Classifying geometries with Cayley. J. Symbolic Comput. 17(3), 259–276 (1994)

    Article  MathSciNet  Google Scholar 

  13. D. Elliott, D. Leemans, E. O’Reilly-Regueiro, Chiral polyhedra and projective lines. Int. J. Math. Stat. 16(1), 45–52 (2015)

    MathSciNet  MATH  Google Scholar 

  14. M.E. Fernandes, D. Leemans, A.I. Weiss, Highly symmetric hypertopes. Aequationes Math. 90, 1045–1067 (2016)

    Article  MathSciNet  Google Scholar 

  15. D. Garbe, Über die Regularen Zerlegungen geschlossener orientierbarer Flächen. J. Reine. Angew. Math. 237, 39–55 (1969)

    MathSciNet  MATH  Google Scholar 

  16. B. Grunbaum, Regularity of graphs, complexes and designs, in Problèmes Combinatoires et Théorie des Graphes, Coll. Internat. CNRS, No. 260, (Orsay, 1977), pp. 191–197

    Google Scholar 

  17. M.I. Hartley, I. Hubard, D. Leemans, Two atlases of abstract chiral polytopes for small groups. Ars Math. Contemp. 5, 371–382 (2012)

    MathSciNet  MATH  Google Scholar 

  18. L. Heffter, Ueber Metacyklische Gruppen und Nachbarcnfigurationem. Math. Ann. 50, 261–268 (1898)

    Article  MathSciNet  Google Scholar 

  19. I. Hubard, D. Leemans, Chiral polytopes and Suzuki simple groups, in Rigidity and Symmetry vol 70, ed. by R. Connelly et al. (Fields Institute Communications, 2014), pp. 155–175

    Google Scholar 

  20. D. Leemans, J. Moerenhout, Chiral polyhedra arising from almost simple groups with socle \(PSL(2, q)\). J. Algebraic Combin. 44, 293–323 (2016)

    Article  MathSciNet  Google Scholar 

  21. D. Leemans, L. Vauthier, An atlas of abstract regular polytopes for small groups. Aequationes Math. 72, 313–320 (2006)

    Article  MathSciNet  Google Scholar 

  22. P. McMullen, E. Schulte, Locally toroidal regular polytopes of rank \(4\). Comment. Math. Helv. 67(1), 77–118 (1992)

    Article  MathSciNet  Google Scholar 

  23. P. McMullen, E. Schulte, in Abstract Regular Polytopes, vol. 92, Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 2002)

    Google Scholar 

  24. P. McMullen, E. Schulte, Locally unitary groups and regular polytopes. Adv. Appl. Math. 29(1), 1–45 (2002)

    Article  MathSciNet  Google Scholar 

  25. B. Monson, A.I. Weiss, Eisenstein integers and related \(C\)-groups. Geom. Dedicata 66, 99–117 (1997)

    Article  MathSciNet  Google Scholar 

  26. B. Nostrand, E. Schulte, Chiral polytopes from hyperbolic honeycombs. Discrete Comput. Geom. 13, 17–39 (1995)

    Article  MathSciNet  Google Scholar 

  27. D. Pellicer, CPR graphs and regular polytopes. European J. Combin. 29(1), 59–71 (2008)

    Article  MathSciNet  Google Scholar 

  28. D. Pellicer, A construction of higher rank chiral polytopes. Discrete Math. 310, 1222–1237 (2010)

    Article  MathSciNet  Google Scholar 

  29. E. Schulte, A.I. Weiss, Chiral polytopes, in Applied Geometry and Discrete Mathematics, volume 4 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci. (American Mathematical Society, Providence, RI, 1991), pp. 493–516

    Google Scholar 

  30. E. Schulte, A.I. Weiss, Chirality and projective linear groups. Discrete Math. 131, 221–261 (1994)

    Article  MathSciNet  Google Scholar 

  31. E. Schulte, A.I. Weiss, Free extensions of chiral polytopes. Canad. J. Math. 47, 641–654 (1995)

    Article  MathSciNet  Google Scholar 

  32. F.A. Sherk, A family of regular maps of type \(\{6, 6\}\). Canad. Math. Bull. 5, 13–20 (1962)

    Article  MathSciNet  Google Scholar 

  33. D.M.Y. Sommerville, An Introduction to the Geometry of n Dimensions (Methuen, London, 1929)

    MATH  Google Scholar 

  34. J. Tits, Sur les analogues algébriques des groupes semi-simples complexes. Colloque d’algébre supérieure, tenu á Bruxelles du 19 au 22 décembre, Centre Belge de Recherches Mathématiques, pp. 261–289, Établissements Ceuterick (Louvain, Librairie Gauthier-Villars, Paris, 1956), p. 1957

    Google Scholar 

  35. T.R.S. Walsh, Hypermaps versus bipartite maps. J. Combinatorial Theory Ser. B 18(2), 155–163 (1975)

    Article  MathSciNet  Google Scholar 

  36. C. Weber, H. Seifert, Die beiden Dodekaederräume. Mat. Z. 37, 237–253 (1933)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank two anonymous referees for their numerous helpful and insightful comments. This research was supported by a Marsden grant (UOA1218) of the Royal Society of New Zealand, by NSERC and by the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), through CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013.

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Authors and Affiliations

  1. Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro, Portugal

    Maria Elisa Fernandes

  2. Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand

    Dimitri Leemans

  3. Département de Mathématique, C.P. 216 - Algèbre et Combinatoire, Université Libre de Bruxelles, Boulevard du Triomphe, 1050, Brussels, Belgium

    Dimitri Leemans

  4. Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada

    Asia Ivić Weiss

Authors
  1. Maria Elisa Fernandes
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  2. Dimitri Leemans
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  3. Asia Ivić Weiss
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Corresponding author

Correspondence to Dimitri Leemans .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Auckland, Auckland, New Zealand

    Marston D. E. Conder

  2. Department of Computing and Software, McMaster University, Hamilton, Ontario, Canada

    Antoine Deza

  3. Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada

    Asia Ivić Weiss

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Fernandes, M.E., Leemans, D., Weiss, A.I. (2018). Hexagonal Extensions of Toroidal Maps and Hypermaps. In: Conder, M., Deza, A., Weiss, A. (eds) Discrete Geometry and Symmetry. GSC 2015. Springer Proceedings in Mathematics & Statistics, vol 234. Springer, Cham. https://doi.org/10.1007/978-3-319-78434-2_8

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