Constructions of Chiral Polytopes of Small Rank

D’Azevedo, Antonio Breda ; Jones, Gareth A. ; Schulte, Egon

Canadian journal of mathematics, Tome 63 (2011) no. 6, pp. 1254-1283

Résumé

An abstract polytope of rank $n$ is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. This paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks 3, 4, and 5.

Détail

DOI : 10.4153/CJM-2011-033-4

Mots-clés : 51M20, 52B15, 05C25, abstract regular polytope, chiral polytope, chiral maps

Bibliographie

[1] [1] Artin, E., Geometric algebra. Interscience Publishers, Inc., New York-London, 1957. Google Scholar

[2] [2] Breda, A., Breda D’Azevedo, A., and Nedela, R., Chirality group and chirality index of the Coxeter chiral maps. Ars Combin. 81(2006), 147–160. Google Scholar

[3] [3] Breda D’Azevedo, A. J. and Jones, G. A., Double coverings and reflexible abelian hypermaps. Beiträge Algebra Geom. 41(2000), no. 2, 371–389. Google Scholar

[4] [4] Breda D’Azevedo, A. J. and Nedela, R., Join and intersection of hypermaps. Acta Univ. M. Belli Ser. Math. 9(2001), 13–28. Google Scholar

[5] [5] Breda D’Azevedo, A. J., Nedela, R., and Širáň, J., Classification of regular maps of negative prime Euler characteristic. Trans. Amer. Math. Soc. 357(2005), no. 10, 4175–4190. doi:10.1090/S0002-9947-04-03622-0 Google Scholar

[6] [6] Breda D’Azevedo, A. J., Jones, G., Nedela, R., and M. Škoviera, Chirality groups of maps and hypermaps. J. Algebraic Combin. 29(2009), no. 3, 337–355. doi:10.1007/s10801-008-0138-z Google Scholar

[7] [7] Conder, M., Hurwitz groups: a brief survey. Bull. Amer. Math. Soc. (N.S.) 23(1990), no. 2, 359–370. doi:10.1090/S0273-0979-1990-15933-6 Google Scholar

[8] [8] Conder, M. D. E., Regular maps and hypermaps of Euler characteristic −1 to −200. J. Combin. Theory Ser. B 99(2009), no. 2, 455–459. (with associated lists of computational data available at http://www.math.auckland.ac.nz/_conder/hypermaps.html.) doi:10.1016/j.jctb.2008.09.003 Google Scholar

[9] [9] Conder, M. D. E., Širáň, J., and Tucker, T.W., The genera, reflexibility and simplicity of regular maps. J. Eur. Math. Soc. 12(2010), no. 2, 343–364. doi:10.4171/JEMS/200 Google Scholar

[10] [10] Conder, M., Hubard, I., and Pisanski, T., Constructions for chiral polytopes. J. London Math. Soc. (2) 77(2008), no. 1, 115–129. doi:10.1112/jlms/jdm093 Google Scholar

[11] [11] Conway, J. H., Curtis, R. T., Norton, S .P., Parker, R. A., and Wilson, R. A., Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. Oxford University Press, Eynsham, 1985. Google Scholar

[12] [12] Coxeter, H. S. M., Twisted honeycombs. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, 4, American Mathematical Society, Providence, RI, 1970. Google Scholar

[13] [13] Coxeter, H. S. M., Regular polytopes. Third ed., Dover Publications, New York, 1973. Google Scholar

[14] [14] Coxeter, H. S. M., Ten toroids and fifty-seven hemidodecahedra. Geom. Dedicata 13(1982), no. 1, 87–99. Google Scholar

[15] [15] Coxeter, H. S. M., A symmetrical arrangement of eleven hemi-icosahedra. In: Convexity and graph theory (Jerusalem 1981), North-HollandMath. Stud., 87, North-Holland, Amsterdam, 1984, pp. 103–114. Google Scholar

[16] [16] Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for discrete groups. Fourth ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, 14, Springer-Verlag, Berlin-New York, 1980. Google Scholar

[17] [17] Coxeter, H. S. M. and Weiss, A. I., Twisted honeycombs ﹛3, 5, 3﹜t and their groups. Geom. Dedicata 17(1984), no. 2, 169–179. Google Scholar

[18] [18] Dickson, L. E., Linear groups: With an exposition of the Galois field theory. Dover Publications Inc., New York, 1958. Google Scholar

[19] [19] Doro, S. and Wilson, S. E., Rotary maps of type ﹛6, 6﹜4. Quarterly J. Math. Oxford Ser. (2) 31(1980), no. 124, 403–414. doi:10.1093/qmath/31.4.403 Google Scholar

[20] [20] The GAP Group, GAP—Groups, Algorithms, and Programming. Version 4.4.9; 2006, http://www.gap-system.org. Google Scholar

[21] [21] Grünbaum, B., Regularity of graphs, complexes and designs. In: Problèmes combinatoires et théorie des graphes (Colloq. Internal. CNRS, Univ. Orsay, Orsay, 1976), Colloq. Internat. CNRS, 260, CNRS, Paris, 1977, pp. 191–197. Google Scholar

[22] [22] Hartley, M. I., All polytopes are quotients, and isomorphic polytopes are quotients by conjugate subgroups. Discrete Comput. Geom. 21(1999), no. 2, 289–298. doi:10.1007/PL00009422 Google Scholar

[23] [23] Hartley, M. I. and Leemans, D., On locally spherical polytopes of type ﹛5, 3, 5﹜. Discrete Math. 309(2009), no. 1, 247–254. doi:10.1016/j.disc.2007.12.084 Google Scholar

[24] [24] Hartley, M. I., McMullen, P., and Schulte, E., Symmetric tessellations on Euclidean space-forms. Canad. J. Math. 51(1999), no. 6, 1230–1239. doi:10.4153/CJM-1999-055-6 Google Scholar

[25] [25] Heffter, L., Über metazyklische Gruppen und Nachbarkonfigurationen. Math. Ann. 50(1898), no. 2–3, 261–268. doi:10.1007/BF01448067 Google Scholar

[26] [26] Hubard, I., A. Orbanıc, and Weiss, A. I., Monodromy groups and self-invariance. Canad. J. Math. 61(2009), no. 6, 1300–1324. doi:10.4153/CJM-2009-061-5 Google Scholar

[27] [27] Hubard, I. and Weiss, A. I., Self-duality of chiral polytopes. J. Combin. Theory Ser. A 111(2005), no. 1, 128–136. doi:10.1016/j.jcta.2004.11.012 Google Scholar

[28] [28] Huppert, B., Endliche Gruppen. I. Die Grundlehren der Mathematischen Wissenschaften, 134, Springer-Verlag, Berlin-New York, 1967. Google Scholar

[29] [29] Huppert, B. and Blackburn, N., Finite groups. III. Grundlehren der Mathematischen Wissenschaften, 243, Springer-Verlag, Berlin-New York, 1982. Google Scholar

[30] [30] Jones, G. A. and Long, C. D., Epimorphic images of the [5, 3, 5] Coxeter group. Math. Z., to appear. Google Scholar

[31] [31] Jones, G. A. and Mednykh, A., Three-dimensional hyperbolic manifolds with a large isometry group. Math. Proc. Cambridge Phil. Soc., to appear. Google Scholar

[32] [32] Jones, G. A. and Silver, S. A., Suzuki groups and surfaces. J. London Math. Soc. (2) 48(1993), no. 1, 117–125. doi:10.1112/jlms/s2-48.1.117 Google Scholar

[33] [33] Jones, G. A. and Singerman, D., Complex functions. An algebraic and geometric viewpoint. Cambridge University Press, Cambridge, 1987. Google Scholar

[34] [34] Jones, G. A. and Singerman, D., Maps, hypermaps and triangle groups. In: The Grothendieck theory of dessins d’enfants (Luminy, 1993), London Math. Soc. Lecture Note Ser., 200, Cambridge University Press, Cambridge, 1994, pp. 115–145. Google Scholar

[35] [35] Jones, G. A., Long, C. D., and Mednykh, A., Hyperbolic manifolds and tessellations of type ﹛3, 5, 3﹜ associated with L2(q). arxiv:1106.0867 Google Scholar

[36] [36] Leemans, D. and Schulte, E., Groups of type L2(q) acting on polytopes. Adv. Geom. 7(2007), no. 4, 529–539. doi:10.1515/ADVGEOM.2007.031 Google Scholar

[37] [37] Long, C. D., Epimorphic images of simplicial Coxeter groups and some associated hyperbolic manifolds, Ph.D. Thesis, University of Southampton, 2007. Google Scholar

[38] [38] Macbeath, A. M., Generators of the linear fractional groups. In: Number theory (Proc. Sympos. Pure Math., Vol. XII, Houston, Texas, 1967), American Mathematical Society, Providence, RI, 1969, pp. 14–32. Google Scholar

[39] [39] McMullen, P., The groups of the regular star-polytopes. Canad. J. Math. 50(1998), no. 2, 426–448. doi:10.4153/CJM-1998-023-7 Google Scholar

[40] [40] McMullen, P. and Schulte, E., Abstract regular polytopes. Encyclopedia of Mathematics and its Applications, 92, Cambridge University Press, Cambridge, 2002. Google Scholar

[41] [41] McMullen, P. and Schulte, E., Hermitian forms and locally toroidal regular polytopes. Adv. Math. 82(1990), no. 1, 88–125. doi:10.1016/0001-8708(90)90084-Z Google Scholar

[42] [42] McMullen, P. and Schulte, E., Regular polytopes of type ﹛4, 4, 3﹜ and ﹛4, 4, 4﹜. Combinatorica 12(1992), no. 2, 203–220. doi:10.1007/BF01204723 Google Scholar

[43] [43] Monson, B. and Schulte, E., Reflection groups and polytopes over finite fields. I. Adv. in Appl. Math. 33(2004), no. 2, 290–317. doi:10.1016/j.aam.2003.11.002 Google Scholar

[44] [44] Monson, B. and Schulte, E., Reflection groups and polytopes over finite fields. II. Adv. in Appl. Math. 38(2007), no. 3, 327–356. doi:10.1016/j.aam.2005.12.001 Google Scholar

[45] [45] Monson, B. and Schulte, E., Modular reduction in abstract polytopes. Canad. Math. Bull. 52(2009), no. 3, 435–450. doi:10.4153/CMB-2009-047-7 Google Scholar

[46] [46] Monson, B. and Weiss, A. I., Regular 4-polytopes related to general orthogonal groups. Mathematika 37(1990), no. 1, 106–118. doi:10.1112/S0025579300012845 Google Scholar

[47] [47] Monson, B. and Weiss, A. I., Eisenstein integers and related C-groups. Geom. Dedicata 66(1997), no. 1, 99–117. doi:10.1023/A:1004953220074 Google Scholar

[48] [48] Nostrand, B. and Schulte, E., Chiral polytopes from hyperbolic honeycombs. Discrete Comput. Geom. 13(1995), no. 2, 17–39. doi:10.1007/BF02574026 Google Scholar

[49] [49] Orbanic, A., F-actions and parallel-product decomposition of reflexible maps. J. Algebraic Combin. 26(2007), no. 4, 507–527. doi:10.1007/s10801-007-0069-0 Google Scholar

[50] [50] Pellicer, D., A construction of higher rank chiral polytopes. Discrete Math. 310(2010), no. 6–7. 1222–1237. doi:10.1016/j.disc.2009.11.034 Google Scholar

[51] [51] Pellicer, D. and Weiss, A. I., Generalized CPR-graphs and applications. Contrib. Discrete Math. 5(2010), no. 2, 76–105. Google Scholar

[52] [52] Ree, R., A family of simple groups associated with the simple Lie algebra of type (G2). Bull. Amer. Math. Soc. 66(1960), 508–510. doi:10.1090/S0002-9904-1960-10523-X Google Scholar

[53] [53] Ree, R., A family of simple groups associated with the simple Lie algebra of type (G2). Amer. J. Math. 83(1961), 432–462. doi:10.2307/2372888 Google Scholar

[54] [54] Schulte, E. and Weiss, A. I., Chiral polytopes. In: Applied geometry and discrete mathematics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 4, American Mathematical Society, Providence, RI, 1991, pp. 493–516. Google Scholar

[55] [55] Schulte, E. and Weiss, A. I., Chirality and projective linear groups. Discrete Math. 131(1994), no. 1–3, 221–261. doi:10.1016/0012-365X(94)90387-5 Google Scholar

[56] [56] Schulte, E. and Weiss, A. I., Free extensions of chiral polytopes. Canad. J. Math. 47(1995), no. 3, 641–654. doi:10.4153/CJM-1995-033-7 Google Scholar

[57] [57] Suzuki, M., On a class of doubly transitive groups. Ann. of Math. 75(1962), 105–145. doi:10.2307/1970423 Google Scholar

[58] [58] Weiss, A. I., Twisted honeycombs ﹛3, 5, 3﹜t . C. R. Math. Rep. Acad. Sci. Canada 5(1983), no. 5, 211–215. Google Scholar

[59] [59] Wilson, S., The smallest nontoroidal chiral maps. J. Graph Theory 2(1978), no. 4, 315–318. doi:10.1002/jgt.3190020405 Google Scholar

[60] [60] Wilson, S., Parallel products in groups and maps. J. Algebra 167(1994), no. 3, 539–546. doi:10.1006/jabr.1994.1200 Google Scholar