Free Extensions of Chiral Polytopes

Schulte, Egon ; Weiss, Asia Ivić

Canadian journal of mathematics, Tome 47 (1995) no. 3, pp. 641-654

Résumé

Abstract polytopes are discrete geometric structures which generalize the classical notion of a convex polytope. Chiral polytopes are those abstract polytopes which have maximal symmetry by rotation, in contrast to the abstract regular polytopes which have maximal symmetry by reflection. Chirality is a fascinating phenomenon which does not occur in the classical theory. The paper proves the following general extension result for chiral polytopes. If K is a chiral polytope with regular facets F then among all chiral polytopes with facets K there is a universal such polytope P, whose group is a certain amalgamated product of the groups of K and F. Finite extensions are also discussed.

Détail

DOI : 10.4153/CJM-1995-033-7

Mots-clés : 51M20, 52A25

Bibliographie

[1] 1. Coxeter, H.S.M., Regular Polytopes, 3rd éd., Dover, New York, 1973. Google Scholar

[2] 2. Coxeter, H.S.M., Twisted Honeycombs, Regional Conf. Ser. in Math., Amer. Math. Soc., Providence, Rhode Island, 1970. Google Scholar

[3] 3. Coxeter, H.S.M., , Regular honeycombs in hyperbolic space, Twelve Geometric Essays, Southern Illinois University Press, Carbondale, 1968. Google Scholar

[4] 4. Coxeter, H.S.M. and Moser, W.O.J., Generators and Relations for Discrete Groups, 4th éd., Springer, Berlin, 1980. Google Scholar

[5] 5. Danzer, L. and Schulte, E., Regulàre Inzidenzkomplexe I, Geom. Dedicata 13(1982), 295–208. Google Scholar

[6] 6. Garbe, D., Uber die regulâren Zerlegungen geschlossener orientierbarer Flâchen, J. Reine Angew. Math. 237(1969), 39–55. Google Scholar

[7] 7. Garbe, D., A remark on nonsymmetric compact Riemann surfaces, Archiv. Math. 30(1978), 435–437. Google Scholar

[8] 8. Griinbaum, B., Regularity of graphs, complexes and designs, Problèmes Combinatoire et Théorie des Graphes, Coll. Internat. CNRS 260, Orsay, 1977. 191–197. Google Scholar

[9] 9. Lyndon, R.C. and Schupp, P.E., Combinatorial Group Theory, Springer, Berlin, 1977. Google Scholar

[10] 10. Magnus, W., Noneuclidean Tesselations and Their Groups, Academic Press, New York, 1974. Google Scholar

[11] 11. Magnus, W., Karrass, A. and Solitar, D., Combinatorial Group Theory, 2nd éd., Dover, New York, 1976. Google Scholar

[12] 12. Malcev, A.I., On the faithful representations of infinite groups by matrices, (Russian) Math. Sb. 8(1940), 405–422. English transi.: Amer. Math. Soc. Transi. (2) 45(1965), 1–18. Google Scholar

[13] 13. McMullen, P., Combinatorially regular polytopes, Mathematika 14(1967), 142–150. Google Scholar

[14] 14. McMullen, P. and Schulte, E., Regular polytopes from twisted Coxeter and unitary reflexion groups, Adv. in Math. 82(1990), 35–87. Google Scholar

[15] 15. McMullen, P., Hermitian forms and locally toroidal regular polytopes, Adv. in Math. 82(1990), 88–125. Google Scholar

[16] 16. McMullen, P., Higher toroidal regular polytopes, Adv. in Math., to appear. Google Scholar

[17] 17. McMullen, P., Quotients of polytopes and C-groups, Discrete Comput. Geom. 11(1994), 453–464. Google Scholar

[18] 18. McMullen, P., Finite quotients of infinite universal polytopes, Discrete Comput. Geom., DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 6(1991), (eds. Goodman, J.E., Pollack, R. and Steiger, W.), 231–236. Google Scholar

[19] 19. McMullen, P., Abstract Regular Polytopes, manuscript in preparation. Google Scholar

[20] 20. Nostrand, B., manuscript in preparation. Google Scholar

[21] 21. Schulte, E., On arranging regular incidence-complexes as faces of higher-dimensional ones, European J. Combin. 4(1983), 375–384. Google Scholar

[22] 22. Schulte, E., Extensions of regular complexes, Finite Geometries, Lecture Notes in Pure and Appl. Math. 103(1985), 289–305. Google Scholar

[23] 23. Schulte, E. and Weiss, A.I., Chiral Polytopes, Applied Geometry and Discrete Mathematics (The “Victor Klee Festschrift“), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 4(1991), (eds. Gritzmann, P. and B.Sturmfels), 493–516. Google Scholar

[24] 24. Schulte, E., Chirality and projective linear groups, Discrete Math. 131(1994), 221–261. Google Scholar

[25] 25. Sherk, F.A., The regular maps on a surface of genus 3, Canad. J. Math. 11(1959), 452–480. Google Scholar

[26] 26. Sherk, F.A., A family of regular maps of type, ﹛6,6﹜, Canad. Math. Bull. 5(1962), 13–20. Google Scholar

[27] 27. Vince, A., Regular combinatorial maps, J. Combin. Theory Ser. B 35(1983), 256–277. Google Scholar

[28] 28. Wehrfritz, B.A., Infinite Linear Groups, Springer-Verlag, New York, 1973. Google Scholar

[29] 29. Weiss, A.I., Incidence-polytopes with toroidal cells, Discrete Comput. Geom. 4(1989), 55–73. Google Scholar

[30] 30. Wilker, J.B., The quaternion formalism for Mobius groups in four or fewer dimensions, Linear Algebra Appl. 19(1993), 99–136. Google Scholar