Abstract
Hypertope is a generalization of the concept of polytope, which in turn generalizes the concept of a map and hypermap, to higher rank objects. Regular hypertopes with spherical residues, which we call regular locally spherical hypertopes, must be either of spherical, euclidean, or hyperbolic type. That is, the type-preserving automorphism group of a locally spherical regular hypertope is a quotient of a finite irreducible, infinite irreducible, or compact hyperbolic Coxeter group. We classify the locally spherical regular hypertopes of spherical and euclidean type and investigate finite hypertopes of hyperbolic type, giving new examples and summarizing some known results.
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Notes
Note that not all symmetries of the cubes appear in that geometry, only half of them as they have to preserve the colors of the vertices.
References
Aschbacher, M.: Flag structures on Tits geometries. Geom. Dedicata 14(1), 21–32 (1983)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997)
Breda D’Azevedo, A., Jones, G.A., Schulte, E.: Constructions of chiral polytopes of small rank. Can. J. Math. 63(6), 1254–1283 (2011)
Buekenhout, F.: Diagrams for geometries and groups. J. Comb. Theory Ser. A 27(2), 121–151 (1979)
Buekenhout, F., Cohen, A.M.: Diagram Geometry. Related to Classical Groups and Buildings. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge. A Series of Modern Surveys in Mathematics, vol. 57. Springer, Heidelberg (2013)
Buekenhout, F., Hermand, M.: On flag-transitive geometries and groups. Travaux de Mathématiques de l’Université Libre de Bruxelles 1, 45–78 (1991)
Cacciari, L.: On the automorphism group of a toroidal hypermap. RAIRO Inform. Théor. Appl. 26(6), 523–540 (1992)
Conder, M.D.E.: Regular maps and hypermaps of Euler characteristic \(-1\) to \(-200\). J. Comb. Theory Ser. B 99(2), 455–459 (2009). Associated lists of computational data at http://www.math.auckland.ac.nz/~conder/hypermaps.html
Conder, M.D.E., Leemans, D., Pellicer, D.: The smallest regular hypertopes (in preparation)
Corn, D., Singerman, D.: Regular hypermaps. Eur. J. Comb. 9(4), 337–351 (1988)
Coxeter, H.S.M.: Configurations and maps. Rep. Math. Colloq. 8, 18–38 (1949)
Coxeter, H.S.M.: Regular Polytopes. Dover, New York (1973)
Danzer, L., Schulte, E.: Reguläre Inzidenzkomplexe I. Geom. Dedicata 13(3), 295–308 (1982)
Ens, E.: Rank 4 toroidal hypertopes. Ars Math. Contemp. 15(1), 67–79 (2018)
Fernandes, M.E., Leemans, D.: C-groups of high rank for the symmetric groups. J. Algebra 508, 196–218 (2018)
Fernandes, M.E., Leemans, D., Ivić Weiss, A.: Highly symmetric hypertopes. Aequationes Math. 90(5), 1045–1067 (2016)
Grünbaum, B.: Regularity of graphs, complexes and designs. In: Problèmes Combinatoires et Théorie des Graphes. Colloq. Internat. CNRS, vol. 260, pp. 191–197. CNRS, Paris (1978)
Hartley, M.I.: An exploration of locally projective polytopes. Combinatorica 28(3), 299–314 (2008)
Hartley, M.I., Leemans, D.: On locally spherical polytopes of type \(\{5,3,5\}\). Discrete Math. 309(1), 247–254 (2009)
Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)
Leemans, D., Schulte, E., Ivić Weiss, A.: Toroidal hypertopes (in preparation)
Maxwell, G.: The normal subgroups of finite and affine Coxeter groups. Proc. Lond. Math. Soc. 76(2), 359–382 (1998)
McMullen, P.: The groups of the regular star-polytopes. Can. J. Math. 50(2), 426–448 (1998)
McMullen, P., Schulte, E.: Higher toroidal regular polytopes. Adv. Math. 117(1), 17–51 (1996)
McMullen, P., Schulte, E.: Abstract Regular Polytopes. Encyclopedia of Mathematics and its Applications, vol. 92. Cambridge University Press, Cambridge (2002)
Monson, B., Schulte, E.: Modular reduction in abstract polytopes. Can. Math. Bull. 52(3), 435–450 (2009)
Schulte, E.: Reguläre Inzidenzkomplexe II. Geom. Dedicata 14(1), 33–56 (1983)
Schulte, E.: Reguläre Inzidenzkomplexe III. Geom. Dedicata 14(1), 57–79 (1983)
Schulte, E.: Amalgamation of regular incidence-polytopes. Proc. Lond. Math. Soc. 56(2), 303–328 (1988)
Tits, J.: Groupes et géométries de Coxeter. In: Jacques Tits, Œuvres Collected Works, vol. 1. Heritage of European Mathematics, pp. 803–817. European Mathematical Society, Zürich (2013)
Acknowledgements
This work is supported by the Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT – Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020, and by NSERC. The authors would also like to thank Gareth Jones for his help with references to maps and hypermaps, and to Antonio Montero for helpful suggestions. Finally, the authors thank an anonymous referee whose numerous comments improved a preliminary version of this paper.
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Fernandes, M.E., Leemans, D. & Weiss, A.I. An Exploration of Locally Spherical Regular Hypertopes. Discrete Comput Geom 64, 519–534 (2020). https://doi.org/10.1007/s00454-020-00209-9
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DOI: https://doi.org/10.1007/s00454-020-00209-9