Advertisement

Aequationes mathematicae

, Volume 90, Issue 5, pp 1045–1067 | Cite as

Highly symmetric hypertopes

  • Maria Elisa Fernandes
  • Dimitri LeemansEmail author
  • Asia Ivić Weiss
Article

Abstract

We study incidence geometries that are thin and residually connected. These geometries generalise abstract polytopes. In this generalised setting, guided by the ideas from the polytope theory, we introduce the concept of chirality, a property of orderly asymmetry occurring frequently in nature as a natural phenomenon. The main result in this paper is that automorphism groups of regular and chiral thin residually connected geometries need to be C-groups in the regular case and \({C^+}\)-groups in the chiral case.

Keywords

Regularity chirality thin geometries hypermaps abstract polytopes 

Mathematics Subject Classification

51E24 52B11 20F05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aschbacher M.: Flag structures on Tits geometries. Geom. Ded. 14(1), 21–32 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bosma W., John J.C., Playoust C.: The magma algebra system I: the user language. J. Symb. Comput. 3/4, 235–265 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D’Azevedo A.B., Jones G.A., Schulte E.: Constructions of chiral polytopes of small rank. Can. J. Math. 63(6), 1254–1283 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D’Azevedo A.B., Jones G.A., Nedela R., Škoviera M.: Chirality groups of maps and hypermaps. J. Algebra Combin. 29, 337–355 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    D’Azevedo A.B., Nedela R.: Chiral hypermaps of small genus. Beiträge Algebra Geom. 44(1), 127–143 (2003)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Buekenhout F.: Diagrams for geometries and groups. J. Combin. Theory Ser. A 27(2), 121–151 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buekenhout, F.: \({(g,\,d^{\ast},\,d)}\)-gons. In: Finite geometries (Pullman, Wash., 1981), volume 82 of Lecture Notes in Pure and Appl. Math., pp. 93–111. Dekker, New York, (1983)Google Scholar
  8. 8.
    Buekenhout, F. (ed.): Handbook of Incidence Geometry: Buildings and Foundations. Elsevier, North-Holland (1995)Google Scholar
  9. 9.
    Buekenhout, F., Cohen, A.M.: 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], 57. Springer, Heidelberg, pp. xiv+592 (2013)Google Scholar
  10. 10.
    Buekenhout, F., Dehon, M., Leemans, D.: An Atlas of residually weakly primitive geometries for small groups. Mém. Cl. Sci., Coll. 8, Ser. 3, Tome XIV. Acad. R. Belgique (1999)Google Scholar
  11. 11.
    Buekenhout F., Hermand M.: On flag-transitive geometries and groups. Travaux de Mathématiques de l’Université Libre de Bruxelles 1, 45–78 (1991)Google Scholar
  12. 12.
    Colbourn C.J., Weiss A.I.: A census of regular 3-polystroma arising from honeycombs. Discr. Math. 50, 29–36 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Conder M., Oliveros D.: The intersection condition for regular polytopes. J. Combin. Theory Ser. A 120(6), 1291–1304 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Connor T., Jambor S., Leemans D.: C-groups of \({PSL(2, q)}\) and \({PGL(2, q)}\). J. Algebra 427, 455–466 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Corn D., Singerman D.: Regular hypermaps. J. Combin. 9, 337–351 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Coxeter, H.S.M., Moser, W.O.J.: 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
  17. 17.
    Coxeter H.S.M., Whitraw G.J.: World-structure and non-Euclidean honeycombs. Proc. R. Soc. Lond. Ser. A 201, 417–437 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Coxeter, H.S.M.: Regular honeycombs in hyperbolic space. Proc. Internat. Congress Math. Amsterdam (1954), Vol. 3; 155–169. North-Holland, Amsterdam (1956)Google Scholar
  19. 19.
    Cunningham G., Pellicer D.: Chiral extensions of chiral polytopes. Discr. Math. 330, 51–60 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Danzer L., Schulte E.: Reguläre Inzidenzkomplexe. I. Geom. Ded. 13(3), 295–308 (1982)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Dehon M.: Classifying geometries with Cayley. J. Symbol. Comput. 17(3), 259–276 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fernandes M.E., Leemans D.: Polytopes of high rank for the symmetric groups. Adv. Math. 228(6), 3207–3222 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fernandes, M.E., Leemans, D., Weiss, A.I.: Hexagonal Extensions of Toroidal Hypermaps. Preprint, p. 19 (2015)Google Scholar
  24. 24.
    Fernandes, M.E., Leemans, D., Weiss, A.I.: Further Extensions of Toroidal Hypermaps. In preparationGoogle Scholar
  25. 25.
    Hartley M.I., Hubard I., Leemans D.: Two atlases of abstract chiral polytopes for small groups. Ars Math. Contemp. 5, 371–382 (2012)MathSciNetzbMATHGoogle Scholar
  26. 26.
    I. HubardWeiss, A.I.: Self-duality of chiral polytopes. J. Combin. Theory Ser. A 111, 128–138 (2005)Google Scholar
  27. 27.
    McMullen, P., Schulte, E.: Abstract Regular Polytopes, volume 92 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2002)Google Scholar
  28. 28.
    Pasini A.: Diagram Geometries. Oxford Science Publications, Oxford (1994)zbMATHGoogle Scholar
  29. 29.
    Pellicer D.: Abstract Chiral Polytopes. Monograph in preparationGoogle Scholar
  30. 30.
    Schulte, E.: Reguläre Inzidenzkomplexe. II, III. Geom. Ded. 14(1), 33–56 and 57–79 (1983)Google Scholar
  31. 31.
    Schulte, E., Weiss, A.I.: Chiral Polytopes. In: Applied geometry and discrete mathematics, volume 4 of DIMACS Ser. Discr. Math. Theor. Comput. Sci., pp. 493–516. Am. Math. Soc., Providence, RI (1991)Google Scholar
  32. 32.
    Tits J.: Les groupes de Lie exceptionnels et leur interprétation géométrique. Bull. Soc. Math. Belg. 8, 48–81 (1956)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Tits, J.: Sur les analogues algébriques des groupes semi-simples complexes. Colloque d’algèbre supérieure, tenu à à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques, 261–289, Établissements Ceuterick, Louvain; Librairie Gauthier-Villars, Paris (1957)Google Scholar
  34. 34.
    Tits J.: Buildings of Spherical Type and Finite BN-Pairs. Springer, Berlin (1974)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Maria Elisa Fernandes
    • 1
  • Dimitri Leemans
    • 2
    Email author
  • Asia Ivić Weiss
    • 3
  1. 1.Department of Mathematics, Center for Research and Development in Mathematics and ApplicationsUniversity of AveiroAveiroPortugal
  2. 2.Department of MathematicsUniversity of AucklandAucklandNew Zealand
  3. 3.Department of Mathematics and StatisticsYork UniversityTorontoCanada

Personalised recommendations