Triangle group representations and constructions of regular maps

Proceedings of the London Mathematical Society

Volumn 82, Issue 3, 2001, Pages 513-532

  Širáň, Jozef ^a  

^a SLOVAK UNIVERSITY OF TECHNOLOGY   (Slovakia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0842337995     PISSN: 00246115     EISSN: None     Source Type: Journal    
DOI: 10.1112/plms/82.3.513     Document Type: Article

References (53)

1. D. ARCHDEACON, P. GVOZDJAK and J. ŠIRÁŇ, 'Constructing and forbidding automorphisms in lifted maps', Math. Slovaca 47 (1997) 113-129.
2. P. BERGAU and D. GARBE, 'Non-orientable and orientable regular maps', Groups - Korea 1988 (ed. A. C. Kim and B. H. Neumann), Lecture Notes in Mathematics 1398 (Springer, Berlin, 1989) 29-42.
3. N. L. BIGGS and A. G. BOSHIER, 'Note on the girth of Ramanujan graphs', J. Combin. Theory Ser. B 49 (1990) 190-194.
4. N. L. BIGGS and A. T. WHITE, Permutation groups and combinatorial structures, London Mathematical Society Lecture Note Series 33 (Cambridge University Press, 1979).
5. H. R. BRAHANA, 'Regular maps and their groups', Amer. J. Math. 49 (1927) 268-284.
6. E. BUJALANCE, 'Normal subgroups of NEC groups', Math. Z. 178 (1981) 331-341.
7. E. BUJALANCE, 'Normal subgroups of even index in an NEC group', Arch. Math. 49 (1987) 470-478.
8. W. BURNSIDE, The theory of groups of finite order (Cambridge University Press, 1911).
9. 2(7)', Math. Proc. Cambridge Philos. Soc. 86 (1979) 395-400.
10. M. D. E. CONDER, 'Generators for alternating and symmetric groups', J. London Math. Soc. (2) 22 (1980) 75-86.
11. M. D. E. CONDER, 'Hurwitz groups - a brief survey', Bull. Amer. Math. Soc. 23 (1990) 359-370.
12. M. D. E. CONDER and R. S. KULKARNI, 'Infinite families of automorphism groups of Riemann surfaces', Discrete groups and geometry (ed. W. J. Harvey and C. Maclachlan), London Mathematical Society Lecture Note Series 173 (Cambridge University Press, 1992) 47-56.
13. M. D. E. CONDER and P. DOBCSÁNYI, 'Determination of all regular maps of small genus', Preprint, University of Auckland, 1999.
14. H. S. M. COXETER and W. O. J. MOSER, Generators and relations for discrete groups, 4th edn (Springer, Berlin, 1984).
15. W. DYCK, 'Über Aufstellung und Untersuchung von Gruppe und Irrationalität regulären Riemannscher Flächen', Math. Ann. 17 (1880) 473-508.
16. C. W. EVANS, 'Net structures and cages', Discrete Math. 27 (1979) 193-204.
17. B. EVERITT, 'Alternating quotients of Fuchsian groups', Preprint, University of Auckland, 1997.
18. D. FRONČEK, A. ROSA and J. ŠIRÁŇ, 'The existence of selfcomplementary circulant graphs', European J. Combin. 17 (1996) 625-628.
19. D. GARBE, 'Über die regulären Zerlegungen geschlossener orientierbarer Flächen', J. Reine Angew. Math. 237 (1969) 39-55.
20. A. GARDINER, R. NEDELA, J. ŠIRÁŇ and M. ŠKOVIERA, 'Characterization of graphs which underlie regular maps on closed surfaces', J. London Math. Soc. (2) 59 (1999) 100-108.
21. J. L. GROSS and T. W. TUCKER, Topological graph theory (Wiley, New York, 1987).
22. B. GRÜNBAUM, 'Regularity of graphs, complexes and designs', Problèmes combinatoires et théorie des graphes, Colloques internationaux C.N.R.S. 260 (Université de Paris-Sud, Orsay, 1976) 191-197.
23. P. GVOZDJAK and J. ŠIRÁŇ, 'Arc-transitive non-Cayley graphs from regular maps', Acta Math. Univ. Comenian. 63 (1994) 309-313.
24. A. HURWITZ, 'Über algebraische Gebilde mit eindeutigen Transformationen in sich', Math. Ann. 41 (1893) 403-442.
25. L. D. JAMES and G. A. JONES, 'Regular orientable imbeddings of complete graphs', J. Combin. Theory Ser. B 39 (1985) 353-367.
26. S. JENDROL, R. NEDELA and M. ŠKOVIERA, 'Constructing regular maps and graphs from planar quotients', Math. Slovaca 47 (1997) 155-170.
27. G. A. JONES, 'Maps on surfaces and Galois groups', Math. Slovaca 47 (1997) 1-33.
28. G. A. JONES and D. SINGERMAN, 'Theory of maps on orientable surfaces', Proc. London Math. Soc. (3) 37 (1978) 273-307.
29. G. A. JONES and D. SINGERMAN, 'Belyǐ functions, hypermaps, and Galois groups', Bull. London Math. Soc. 28 (1996) 561-590.
30. F. KLEIN, 'Über die Transformation siebenter Ordnung der elliptischen Funktionen', Math. Ann. 14 (1879) 428-471.
31. A. W. KNAPP, 'Doubly generated Fuchsian groups', Michigan Math. J. 15 (1969) 289-304.
32. A. LUBOTZKY, R. PHILLIPS and P. SARNAK, 'Ramanujan graphs', Combinatorica 8 (1988) 261-277.
33. A. M. MACBEATH, 'Generators of linear fractional groups', Proceedings of Symposia in Pure Mathematics XII (American Mathematical Society, Providence, RI, 1969) 14-32.
34. W. MAGNUS, Non-euclidean tesselations and their groups (Academic Press, New York, 1974).
35. I. MAL'CEV, 'On the faithful representation of infinite groups by matrices', Mat. Sb. 8 (50) (1940) 405-422 (Russian),
36. Amer. Math. Soc. Transi. (2) 45 (1965) 1-18 (English).
37. G. A. MARGULIS, 'Explicit group-theoretic constructions of combinatorial schemes and their applications in the constructions of expanders and concentrators', Problemy Peredachi Informatsii 24 (1988) 51-60 (Russian),
38. Problems Inform. Transmission 24 (1988) 39-46 (English).
39. B. D. MCKAY and C. E. PRAEGER, 'Vertex-transitive graphs which are not Cayley graphs. I', J. Austral. Math. Soc. Ser. A 56 (1994) 53-63.
40. B. D. MCKAY and C. E. PRAEGER, 'Vertex-transitive graphs that are not Cayley graphs. II', J. Graph Theory 22 (1996) 321-334.
41. J. MENNICKE, 'Eine Bemerkung über Fuchssche Gruppen', Invent. Math. 2 (1967) 301-305.
42. B. MOHAR, 'Face width of embedded graphs', Math. Slovaca 47 (1997) 35-63.
43. J. FLAGG MORGAN, 'The growth rate and balance of homogeneous tilings in the hyperbolic plane', Discrete Math. 173 (1997) 151-186.
44. R. NEDELA and M. ŠKOVIERA, 'Regular maps on surfaces with large planar width', European J. Combin. to appear.
45. R. B. RICHTER, J. ŠIRÁŇ, R. JAJCAY, T. W. TUCKER and M. WATKINS, 'Cayley maps', Preprint, University of Waterloo, Canada, 2000.
46. T. J. RIVLIN, Chebyshev polynomials, from approximation theory to algebra and number theory, 2nd edn (Wiley-Interscience, New York, 1990).
47. N. ROBERTSON and P. D. SEYMOUR, 'Graph minors, VII. Disjoint paths on a surface', J. Combin. Theory Ser. B 45 (1988) 212-254.
48. C. H. SAH, 'Groups related to compact Riemann surfaces', Acta. Math. 123 (1969) 13-42.
49. D. SUROWSKI, 'Lifting map automorphisms and Macbeath's theorem', J. Combin. Theory Ser. B 50 (1990) 135-149.
50. C. THOMASSEN, 'Embeddings of graphs with no short noncontractible cycles', J. Combin. Theory Ser. B 48 (1990) 155-177.
51. A. VINCE, 'Regular combinatorial maps', J. Combin. Theory Ser. B 35 (1983) 256-277.
52. P.-K. WONG, 'Cages - a survey', J. Graph Theory 6 (1982) 1-22.
53. H. ZIESCHANG, E. VOGT and H.-D. COLDEWEY, Surfaces and planar discontinuous groups, Lecture Notes in Mathematics 835 (Springer, Berlin, 1980).