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Classification of regular maps of negative prime Euler characteristic

Authors: Antonio Breda d'Azevedo, Roman Nedela and Jozef Sirán
Journal: Trans. Amer. Math. Soc. 357 (2005), 4175-4190
MSC (2000): Primary 05C10; Secondary 57M15, 57M60, 20F65, 05C25
Published electronically: November 4, 2004
MathSciNet review: 2159705
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Abstract: We give a classification of all regular maps on nonorientable surfaces with a negative odd prime Euler characteristic (equivalently, on nonorientable surfaces of genus $p+2$ where $p$is an odd prime). A consequence of our classification is that there are no regular maps on nonorientable surfaces of genus $p+2$where $p$ is a prime such that $p\equiv 1$ (mod $12$) and $p\ne 13$.

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  • 1. R. D. M. Accola, On the number of automorphisms of a closed Riemann surfaces, Trans. Amer. Math. Soc. 131 (1968), 398-408. MR 36:5333
  • 2. M. Belolipetsky and G. Jones, Automorphism groups of Riemann surfaces of genus $p+1$, where $p$ is a prime, submitted.
  • 3. H. Bender, Finite groups with dihedral Sylow $2$-subgroups, J. Algebra 70 (1981), 216-228.MR 83c:20011b
  • 4. H. Bender and G. Glauberman, Characters of finite groups with dihedral Sylow $2$-subgroups, J. Algebra 70 (1981), 200-215. MR 83c:20011a
  • 5. P. Bergau and D. Garbe, Non-orientable and orientable regular maps, in: Proceedings of ``Groups-Korea 1988", Lect. Notes Math. 1398, Springer (1989), 29-42. MR 90k:57003
  • 6. H. R. Brahana, Regular maps and their groups, Amer. J. Math. 49 (1927), 268-284.
  • 7. R. P. Bryant, D. Singerman, Foundations of the theory of maps on surfaces with boundary, Quart. J. Math. Oxford Ser. (2) 36 (1985), no. 141, 17-41.MR 86f:57008
  • 8. W. Burnside, ``Theory of Groups of Finite Order'', Cambridge Univ. Press, 1911.
  • 9. M. Conder and P. Dobcsányi, Determination of all regular maps of small genus, J. Combinat. Theory Ser. B 81 (2001), 224-242.MR 2002f:05088
  • 10. M. Conder and B. Everitt, Regular maps on non-orientable surfaces, Geom. Dedicata 56 (1995), 209-219. MR 96g:05046
  • 11. H. S. M. Coxeter and W. O. J. Moser, ``Generators and Relations for Discrete Groups", 4th Ed., Springer-Verlag, Berlin, 1984. MR 81a:20001
  • 12. E. Dickson, ``Linear groups with an exposition of Galois field theory'', 1901; Dover Publ., 1958. MR 21:3488
  • 13. W. Dyck, Über Aufstellung und Untersuchung von Gruppe und Irrationalität regularer Riemannscher Flächen, Math. Ann. 17 (1880), 473-508.
  • 14. D. Garbe, Über die regulären Zerlegungen geschlossener orientierbarer Flächen, J. Reine Angew. Math. 237 (1969), 39-55.MR 39:7502
  • 15. A. Gardiner, R. Nedela, J. Sirán and M. Skoviera, Characterization of graphs which underlie regular maps on closed surfaces, J. London Math. Soc. (2) 59 (1999) No. 1, 100-108. MR 2000a:05104
  • 16. D. Gorenstein and 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. MR 31:1297a; MR 31:1297b; MR 32:7634
  • 17. A. Grothendieck, ``Esquisse d'un programme'', Geometric Galois actions, London Math. Soc. Lecture Note Ser., No. 242, Cambridge Univ. Press, Cambridge, 1997, pp. 1, 5-48.MR 99c:14034
  • 18. L. Heffter, Über metazyklische Gruppen und Nachbarconfigurationen, Math. Ann. 50 (1898), 261-268.
  • 19. N. Ito, Über das Produkt von zwei abelschen Gruppen, Math. Z. 62 (1955), 400-401. MR 17:125b
  • 20. L. D. James and G. A. Jones, Regular orientable imbeddings of complete graphs, J. Combinat. Theory Ser. B 39 (1985), 353-367.MR 87a:05060
  • 21. G. A. Jones, Maps on surfaces and Galois groups, Math. Slovaca 47 (1997), 1-33. MR 98i:05055
  • 22. G. A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978), 273-307. MR 58:21744
  • 23. G. A. Jones and D. Singerman, Bely{\u{\i}}\kern.15emfunctions, hypermaps, and Galois groups, Bull. London Math. Soc. 28 (1996), 561-590.MR 97g:11067
  • 24. J. Kepler, ``The harmony of the world'' (translation from the Latin ``Harmonice Mundi'', 1619), Memoirs Amer. Philos. Soc. 209, American Philosophical Society, Philadelphia, PA, 1997. MR 2000c:01020
  • 25. F. Klein, Über die Transformation siebenter Ordnung der elliptischen Functionen, Math. Ann. 14 (1879), 428-471.
  • 26. C. MacLachlan, A bound for the number of automorphisms of a compact Riemann surface, J. London Math. Soc. 44 (1969), 265-272. MR 38:4674
  • 27. C. L. May and J. Zimmerman, There is a group of every strong symmetric genus, Bull. London Math. Soc. 35 (2003), 433-439.MR 2004b:57026
  • 28. R. Nedela and M. Skoviera, Exponents of orientable maps, Proc. London Math. Soc. (3) 75 (1997), 1-31. MR 98i:05059
  • 29. C. H. Sah, Groups related to compact Riemann surfaces, Acta Math. 123 (1969), 13-42. MR 40:4447
  • 30. F. A. Sherk, The regular maps on a surface of genus three, Canad. J. Math. 11 (1959), 452-480. MR 21:5928
  • 31. T. W. Tucker, Finite groups acting on surfaces and the genus of a group, J. Combinat. Theory Ser. B 34 (1983) No. 1, 82-98. MR 85b:20055
  • 32. W. T. Tutte, What is a map?, in ``New Directions in Graph Theory'' (F. Harary, Ed.), Acad. Press, 1973, 309-325. MR 51:12589
  • 33. S. Wilson and A. Breda D'Azevedo, Surfaces with no regular hypermaps, Discrete Math. 277 (2004), 241-274.

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Additional Information

Antonio Breda d'Azevedo
Affiliation: Departamento de Matematica, Universidade de Aveiro, Aveiro, Portugal

Roman Nedela
Affiliation: Institute of Mathematics, Slovak Academy of Science, Banská Bystrica, Slovakia

Jozef Sirán
Affiliation: Department of Mathematics, SvF, Slovak Univ. of Technology, Bratislava, Slovakia

Keywords: Regular maps, nonorientable surfaces, quotients of triangle groups, prime Euler characteristic
Received by editor(s): April 9, 2003
Received by editor(s) in revised form: December 11, 2003
Published electronically: November 4, 2004
Additional Notes: The authors thank the Department of Mathematics of the University of Aveiro and the Research Unit “Matemática e Aplicações” for supporting this project.
The second author acknowledges support from the VEGA Grant No. 2/2060/22 and from the APVT Grant No. 51-012502.
The third author was sponsored by the U.S.-Slovak Science and Technology Joint Fund under Project Number 020/2001, and also in part by the VEGA Grant No. 1/9176/02 and the APVT Grant No. 20-023302.
Article copyright: © Copyright 2004 American Mathematical Society

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