On External Symmetry Groups of Regular Maps

  • Marston D. E. Conder
  • Young Soo Kwon
  • Jozef ŠiráňEmail author
Part of the Fields Institute Communications book series (FIC, volume 70)


Regular maps are embeddings of graphs or multigraphs on closed surfaces (which may be orientable or non-orientable), in which the automorphism group of the embedding acts regularly on flags. Such maps may admit external symmetries that are not automorphisms of the embedding, but correspond to combinations of well known operators that may transform the map into an isomorphic copy: duality, Petrie duality, and the ‘hole operators’, also known as ‘taking exponents’. The group generated by the external symmetries admitted by a regular map is the external symmetry group of the map. We will be interested in external symmetry groups of regular maps in the case when the map admits both the above dualities (that is, if it has trinity symmetry) and all feasible hole operators (that is, if it is kaleidoscopic). Existence of finite kaleidoscopic regular maps was conjectured for every even valency by Wilson, and proved by Archdeacon, Conder and Širáň (2010).

It is well known that regular maps may be identified with quotients of extended triangle groups. In other words, these groups may be regarded as ‘universal’ for constructions of regular maps. It is therefore interesting to ask if similar ‘universal’ groups exist for kaleidoscopic regular maps with trinity symmetry. A satisfactory answer, however, is likely to be very complex, if indeed feasible at all. We demonstrate this (and other things) by a construction of an infinite family of finite kaleidoscopic regular maps with trinity symmetry, all of valency 8, such that the orders of their external symmetry groups are unbounded. Also we explicitly determine the external symmetry groups for the family of kaleidoscopic regular maps of even valency mentioned above.


Regular map Group of external symmetries 

Subject Classifications

05E18 20B25 57M15 



The first author is grateful for support by the Marsden Fund (Grant No. UOA 1015) from the Royal Society of New Zealand. The third author acknowledges support by the VEGA Research Grant No. 1/0781/11 and the APVV Research Grant No. 0223-10. Also the first and third authors are members of the EuroCoREs programme EUROGIGA (project GREGAS, ESF-EC-0009-10) of the European Science Foundation, and the third author’s research is partially supported by that through the APVV.

A substantial part of this research was undertaken while the second and third authors were visiting the first author at the University of Auckland in early 2011, and these authors acknowledge the warm hospitality of the Mathematics Department there.


  1. 1.
    Archdeacon, D.S., Conder, M.D.E., Širáň, J.: Trinity symmetry and kaleidoscopic regular maps. Trans. Amer. Math. Soc. (to appear)Google Scholar
  2. 2.
    Bosma, W., Cannon, C., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comput. 24, 235–265 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bryant, R.P., Singerman, D.: Foundations of the theory of maps on surfaces with boundary. Quart. J. Math. (Oxford) Ser. (2) 36(141), 17–41 (1985)Google Scholar
  4. 4.
    Conder, M.D.E.: On the group G 6, 6, 6. Quart. J. Math. (Oxford) Ser. (2) 39, 175–183 (1988)Google Scholar
  5. 5.
    Conder, M.D.E.: Regular maps and hypermaps of Euler characteristic − 1 to − 200. J. Comb. Theory Ser. B 99, 455–459 (2009). With associated lists of computational data available at
  6. 6.
    Coxeter, H.S.M.: The abstract groups G m, n, p. Trans. Amer. Math. Soc. 45(1), 73–150 (1939)MathSciNetGoogle Scholar
  7. 7.
    Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups, 4th edn. Springer, Berlin (1984)Google Scholar
  8. 8.
    Edjvet, M., Juhász, A.: The groups G m, n, p. J. Algebra 319, 248–266 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Jones, G.A., Singerman, D.: Belyj functions, hypermaps, and Galois groups. Bull. Lond. Math. Soc. 28, 561–590 (1996)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Jones, G.A., Thornton, J.S.: Operations on maps, and outer automorphisms. J. Comb. Theory Ser. B 35(2), 93–103 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Nedela, R., Škoviera, M.: Exponents of orientable maps. Proc. Lond. Math. Soc. (3) 75, 1–31 (1997)Google Scholar
  12. 12.
    Richter, R.B., Širáň, J., Wang, Y.: Self-dual and self-Petrie dual regular maps. J. Graph Theory 69(2), 152–159 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Širáň, J.: Regular maps on a given surface: a survey. In: R. Thomas et al. (ed.) Topics in Discrete Mathematics. Springer Series No. 26 Algorithms and Combinatorics, pp. 591–609. Springer, Berlin (2006)Google Scholar
  14. 14.
    Širáň, J., Wang, J.: Maps with highest level of symmetry that are even more symmetric than other such maps: regular maps with largest exponent groups. Contemporary Mathematics. AMS Ser. 531(Combinatorics and Graphs), 95–102 (2010)Google Scholar
  15. 15.
    Wilson, J.: New techniques for the construction of regular maps. PhD thesis, University of Washington (1976)Google Scholar
  16. 16.
    Wilson, S.E.: Operators over regular maps. Pac. J. Math. 81, 559–568 (1979)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Marston D. E. Conder
    • 1
  • Young Soo Kwon
    • 2
  • Jozef Širáň
    • 3
    Email author
  1. 1.University of AucklandAucklandNew Zealand
  2. 2.Yeungnam UniversityGyeongsanKorea
  3. 3.Open University, UK and Solvak University of TechnologyBratislavaSlovakia

Personalised recommendations