Geodesic
REFLEXIBLE REGULAR MAPS WITH NO NON-TRIVIAL EXPONENTS FROM RESIDUAL FINITENESS
Glasgow mathematical journal, Tome 53 (2011) no. 3, pp. 437-441

Voir la notice de l'article provenant de la source Cambridge University Press

Regular maps, that is, graph embeddings with the ‘highest level’ of orientation-preserving symmetry, can be identified with two-generator presentations of groups G of the form 〈x, y; xm = y2 = (xy)n = ... = 1〉; the parameters m and n are the valence and the covalence of the map, respectively. The element j ∈ Zm* is an exponent of such a map if the assignments x ↦ xj and y ↦ y extend to an automorphism of G. The element −1 is an exponent if and only if the map is reflexible, that is, isomorphic to its mirror image. Non-trivial exponents, j ≠ ±1, induce automorphisms of the underlying graph, which are not map automorphisms but can nevertheless be considered as ‘external symmetries’ of the map. In this paper we show with the help of residual finiteness of triangle groups that for any given m, n≥3 such that 1/m + 1/n ≤ 1/2, there exist infinitely many finite, reflexible and regular maps of valence m and covalence n with no non-trivial exponent.
DOI : 10.1017/S0017089511000048
Mots-clés : 05C25 
ŠIRÁŇ, JOZEF; STANEKOVÁ, ĽUBICA; OLEJÁR, MARIÁN. REFLEXIBLE REGULAR MAPS WITH NO NON-TRIVIAL EXPONENTS FROM RESIDUAL FINITENESS. Glasgow mathematical journal, Tome 53 (2011) no. 3, pp. 437-441. doi: 10.1017/S0017089511000048
@article{10_1017_S0017089511000048,
     author = {\v{S}IR\'A\v{N}, JOZEF and STANEKOV\'A, \v{L}UBICA and OLEJ\'AR, MARI\'AN},
     title = {REFLEXIBLE {REGULAR} {MAPS} {WITH} {NO} {NON-TRIVIAL} {EXPONENTS} {FROM} {RESIDUAL} {FINITENESS}},
     journal = {Glasgow mathematical journal},
     pages = {437--441},
     year = {2011},
     volume = {53},
     number = {3},
     doi = {10.1017/S0017089511000048},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089511000048/}
}
TY  - JOUR
AU  - ŠIRÁŇ, JOZEF
AU  - STANEKOVÁ, ĽUBICA
AU  - OLEJÁR, MARIÁN
TI  - REFLEXIBLE REGULAR MAPS WITH NO NON-TRIVIAL EXPONENTS FROM RESIDUAL FINITENESS
JO  - Glasgow mathematical journal
PY  - 2011
SP  - 437
EP  - 441
VL  - 53
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089511000048/
DO  - 10.1017/S0017089511000048
ID  - 10_1017_S0017089511000048
ER  - 
%0 Journal Article
%A ŠIRÁŇ, JOZEF
%A STANEKOVÁ, ĽUBICA
%A OLEJÁR, MARIÁN
%T REFLEXIBLE REGULAR MAPS WITH NO NON-TRIVIAL EXPONENTS FROM RESIDUAL FINITENESS
%J Glasgow mathematical journal
%D 2011
%P 437-441
%V 53
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089511000048/
%R 10.1017/S0017089511000048
%F 10_1017_S0017089511000048

[1] 1.Archdeacon, D., Gvozdjak, P. and Širáň, J., Constructing and forbidding automorphisms in lifted maps, Mainath. Slovaca 47 (1997), 113–129. Google Scholar

[2] 2.Gardiner, A., Nedela, R., Škoviera, M. and Širáň, J., Characterization of graphs which underlie regular maps on closed surfaces, J. Lond. Math. Soc. (2) 59 (1), (1999), 100–108. Google Scholar | DOI

[3] 3.Jones, G. A. and Singerman, D., Theory of maps on orientable surfaces, Proc. Lond. Math. Soc. 37 (3) (1978), 273–307. Google Scholar | DOI

[4] 4.Maltsev, I., On the faithful representation of infinite groups by matrices, Mat. Sb. 8 (50) (1940) 405–422 (Russian), Amer. Math. Soc. Transl. (2) (1965), 1–18 (English). Google Scholar

[5] 5.Nedela, R. and Škoviera, M., Exponents of orientable maps, Proc. Lond. Math. Soc. 75 (3) (1997), 1–31. Google Scholar | DOI

[6] 6.Richter, R. B., Širáň, J., Jajcay, R., Tucker, T. W. and Watkins, M. E., Cayley maps, J. Comb. Theory Ser. B 95 (2005), 189–245. Google Scholar | DOI

[7] 7.Širáň, J., Regular maps on a given surface: A survey, in Topics in discrete mathematics (Thomas, R. et al. , Editors), Springer series no. 26: Algorithms and combinatoric (Springer, New Mexico, 2006), 591–609. Google Scholar | DOI

[8] 8.Širáň, J. and Wang, Y., Maps with highest level of symmetry that are even more symmetric than other such maps: Regular maps with largest exponent groups, Contemporary Math. (AMS series) 531 (2010), 95–102. Google Scholar | DOI

[9] 9.Wilson, S. E., Operators over regular maps, Pacific J. Math. 81 (1979), 559–568. Google Scholar | DOI

Cité par Sources :