Regular Maps on a Given Surface: A Survey

  • Jozef Širáň
Part of the Algorithms and Combinatorics book series (AC, volume 26)


Regular maps are cellular decompositions of closed surfaces with the highest ‘level of symmetry’, meaning that the automorphism group of the map acts regularly on flags. We survey the state-of-the-art of the problem of classification of regular maps on a given surface and outline directions of future research in this area.


Graph surface embedding map automorphism regular map 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BeJ04]
    M. Belolipetsky, G. A. Jones, Automorphism groups of Riemann surfaces of genus p+1, where p is a prime, Submitted (2004).Google Scholar
  2. [Ben81]
    H. Bender, Finite groups with dihedral Sylow 2-subgroups, J. of Algebra 70 (1981), 216–228.MATHCrossRefMathSciNetGoogle Scholar
  3. [BG181]
    H. Bender, G. Glauberman, Characters of finite groups with dihedral Sylow 2-subgroups, J. Algebra 70 (1981), 200–215.MATHCrossRefMathSciNetGoogle Scholar
  4. [BeG89]
    P. Bergau, D. Garbe, Non-orientable and orientable regular maps, in Proceedings of “Groups-Korea 1988”, Lect. Notes Math. 1398, Springer (1989), 29–42.MathSciNetGoogle Scholar
  5. [Big71]
    N. L. Biggs, Automorphisms of imbedded graphs, J. Combinat. Theory Ser. B 11 (1971), 132–138.MATHCrossRefMathSciNetGoogle Scholar
  6. [Bra26]
    H. R. Brahana, Regular maps on an anchor ring, Amer. J. Math. 48 (1926), 225–240.CrossRefMathSciNetMATHGoogle Scholar
  7. [Bra27]
    H. R. Brahana, Regular maps and their groups, Amer. J. Math. 49 (1927), 268–284.CrossRefMathSciNetMATHGoogle Scholar
  8. [BJN05]
    A. Breda d’Azevedo, G. A. Jones, R. Nedela, M. Škoviera, Chirality groups of maps and hypermaps, submitted (2005).Google Scholar
  9. [BNS05]
    A. Breda d’Azevedo, R. Nedela, J. Širáň, Classification of regular maps of negative prime Euler characteristic, Trans. Amer. Math. Soc. 357 No. 10 (2005), 4175–4190.MATHCrossRefMathSciNetGoogle Scholar
  10. [BrS85]
    R. P. Bryant, D. Singerman, Foundations of the theory of maps on surfaces with boundary, Quart. J. Math. Oxford Ser. (2) 36 No. 141 (1985), 17–41.MATHCrossRefMathSciNetGoogle Scholar
  11. [Burll]
    W. Burnside, Theory of Groups of Finite Order, Cambridge Univ. Press, 1911.Google Scholar
  12. [CoD0l]
    M. D. E. Conder, P. Dobcsányi, Determination of all regular maps of small genus, J. Combinat. Theory Ser. B 81 (2001), 224–242.MATHCrossRefGoogle Scholar
  13. [CoE95]
    M.D.E. Conder, B. Everitt, Regular maps on non-orientable surfaces, Geom. Dedicata 56 (1995), 209–219.MATHCrossRefMathSciNetGoogle Scholar
  14. [CPS04]
    M. D. E. Conder, P. Potočnik, J. Širáň, Regular maps on groups with cyclic odd-Sylow subgroups, preprint (2004).Google Scholar
  15. [CPS05]
    M. D. E. Conder, P. Potočnik, J. Širáň, Regular hypermaps over projective linear groups, preprint (2005).Google Scholar
  16. [Cox48]
    H. S. M. Coxeter, Configurations and maps, Rep. Math. Colloq (2) 8 (1948), 18–38.MathSciNetGoogle Scholar
  17. [CMo84]
    H. S. M. Coxeter, W. O.J. Moser, Generators and Relations for Discrete Groups, 4th Ed., Springer-Verlag, Berlin 1984.Google Scholar
  18. [DKN04]
    S.F. Du, J. H. Kwak, R. Nedela, Regular embeddings of complete multipartite graphs, submitted (2004).Google Scholar
  19. [DKN05]
    S.F. Du, J. H. Kwak, R. Nedela, Classification of regular embeddings of hypercubes of odd dimension, submitted (2005).Google Scholar
  20. [Dyc80]
    W. Dyck, Über Aufstellung und Untersuchung von Gruppe und Irra-tionalität regularer Riemannscher Flächen, Math. Ann. 17 (1880), 473–508.CrossRefMathSciNetGoogle Scholar
  21. [Gar69]
    D. Garbe, Über die regulären Zerlegungen geschlossener orientierbarer Flächen, J. Reine Angew. Math. 237 (1969), 39–55.MATHMathSciNetGoogle Scholar
  22. [Gar78]
    D. Garbe, A remark on non-symmetric Riemann surfaces, Arch. Math. 30 (1978), 435–437.MATHCrossRefMathSciNetGoogle Scholar
  23. [GNS99]
    A. Gardiner, R. Nedela, J. Širáň, M. Škoviera, Characterization of graphs which underlie regular maps on closed surfaces, J. London Math. Soc. (2) 59 No. 1 (1999), 100–108.MATHCrossRefMathSciNetGoogle Scholar
  24. [GoW65]
    D. Gorenstein, J.H. Walter, The characterization of finite groups with dihedral Sylow 2-subgroups, I, II, III, J. of Algebra 2 (1965), 85–151, 218–270, 334–393.CrossRefMathSciNetGoogle Scholar
  25. [Gre63]
    A. S. Grek, Reqular polyhedra of simplest hyperbolic types, (Russian) Ivanov. Gos. Ped. Inst. Učen. Zap. 34 (1963), 27–30.MathSciNetGoogle Scholar
  26. [Gr66a]
    A. S. Grek, Regular polyhedrons on surfaces with Euler characteristic χ = −4, (Russian) Soobšč. Akad. Nauk Gruzin. SSR 42 (1966), 11–15.MATHMathSciNetGoogle Scholar
  27. [Gr66b]
    A. S. Grek, Regular polyhedra on a closed surface with the Euler characteristic χ = −3, (Russian) Izv. Vysš. Učebn. Zaved. Matematika 55 No. 6 (1966), 50–53.MathSciNetGoogle Scholar
  28. [Gro84]
    A. Grothendieck, Esquisse d’un programme, preprint, Montpellier (1984).Google Scholar
  29. [GrT87]
    J.L. Gross, T.W. Tucker, Topological Graph Theory, Wiley (1987) and Dover (2001).Google Scholar
  30. [Hea90]
    P. J. Heawood, Map-colour theorem. Quart, J. Math. 24 (1890), 332–338.Google Scholar
  31. [Hef98]
    L. Heffter, Über metazyklische Gruppen und Nachbarconfigurationen, Math. Ann. 50 (1898), 261–268.CrossRefMathSciNetMATHGoogle Scholar
  32. [JaJ85]
    L.D. James, G. A. Jones, Regular orientable imbeddings of complete graphs, J. Combinat. Theory Ser. B 39 (1985), 353–367.MATHCrossRefMathSciNetGoogle Scholar
  33. [Jon97]
    G. A. Jones, Maps on surfaces and Galois groups, Math. Slovaca 47 (1997), 1–33.MATHMathSciNetGoogle Scholar
  34. [Jon04]
    G. A. Jones, Large groups of automorphisms of compact surfaces with prime negative Euler characteristic, preprint (2004).Google Scholar
  35. [JNS05]
    G. A. Jones, R. Nedela, J. Širáň, Classification of regular maps of Euler characteristic −3p, where p is a prime, preprint (2005).Google Scholar
  36. [JNS04]
    G. A. Jones, R. Nedela, M. Škoviera, Regular embeddings of K n,n where n is odd prime power, preprint (2004).Google Scholar
  37. [JoS78]
    G. A. Jones, D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978), 273–307.MATHCrossRefMathSciNetGoogle Scholar
  38. [JoS96]
    G. A. Jones, D. Singerman, Belyĭ functions, hypermaps, and Galois groups, Bull. London Math. Soc. 28 (1996), 561–590.CrossRefMathSciNetGoogle Scholar
  39. [Kepl9]
    J. Kepler, The harmony of the world, (Translation from the Latin “Har-monice Mundi”, 1619) Memoirs Amer. Philos. Soc. 209, American Philosophical Society, Philadelphia, PA (1997).Google Scholar
  40. [Kle79]
    F. Klein, Über die Transformation siebenter Ordnung der elliptischen Func-tionen, Math. Ann. 14 (1879), 428–471.CrossRefGoogle Scholar
  41. [KwK05]
    J. H. Kwak, Y. S. Kwon, Regular orientable embeddings of complete bipartite graphs, preprint (2005).Google Scholar
  42. [Sah69]
    C.H. Sah, Groups related to compact Riemann surfaces, Acta Math. 123 (1969), 13–42.MATHCrossRefMathSciNetGoogle Scholar
  43. [Sch85]
    J. Scherwa, Regulaere Karten geschlossener nichtorientierbarer Flaechen, Diploma Thesis, Bielefeld (1985).Google Scholar
  44. [She59]
    F. A. Sherk, The regular maps on a surface of genus three, Ganad. J. Math. 11 (1959), 452–480.MATHMathSciNetGoogle Scholar
  45. [Suz55]
    M. Suzuki, On finite groups with cyclic Sylow subgroups for all odd primes, Amer. J. Math. 77 (1955), 657–691.MATHCrossRefMathSciNetGoogle Scholar
  46. [Tuc83]
    T. W. Tucker, Finite groups acting on surfaces and the genus of a group, J. Gombinat. Theory Ser. B 34 No. 1 (1983), 82–98.MATHCrossRefMathSciNetGoogle Scholar
  47. [Tut73]
    W. T. Tutte, What is a map? In New Directions in Graph Theory (F. Harary, Ed.), Acad. Press (1973), 309–325.Google Scholar
  48. [Wi78a]
    S. E. Wilson, Non-orientable regular maps, Ars Gombin. 5 (1978), 213–218.MATHGoogle Scholar
  49. [Wi78b]
    S.E. Wilson, Riemann surfaces over regular maps, Canad. J. Math. 30 (1978), 763–782.MATHMathSciNetGoogle Scholar
  50. [WBr04]
    S. E. Wilson, A. Breda D’Azevedo, Surfaces having no regular hypermaps, Discrete Math. 277 (2004), 241–274.MATHCrossRefMathSciNetGoogle Scholar
  51. [Won66]
    W. J. Wong, On finite groups with semi-dihedral Sylow 2-subgroups, J. Algebra 4 (1966), 52–63.CrossRefMathSciNetGoogle Scholar
  52. [Zas36]
    H. Zassenhaus, Über endliche Fastkörper, Abh. Math. Sem. Univ. Hamburg 11 (1936), 187–220.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jozef Širáň
    • 1
    • 2
  1. 1.Department of Mathematics, SvFSlovak University of TechnologyBratislavaSlovak Republic
  2. 2.Department of MathematicsUniversity of AucklandAucklandNew Zealand

Personalised recommendations