Abstract
For each \(d\ge 3\), \(n \ge 5\), and \(k_1, k_2, \ldots , k_{d-1}\ge 2\) with \(k_1+k_2+\cdots +k_{d-1}\le n-1\), we show how to construct a regular d-polytope whose automorphism group is of order \(2^n\) and whose Schläfli type is \(\{2^{k_1},2^{k_2}, \ldots , 2^{k_{d-1}}\}\).
Similar content being viewed by others
References
Besche, H.U., Eick, B., O’Brien, E.A.: The groups of order at most 2000. Electron. Res. Announc. Am. Math. Soc. 7, 1–4 (2001)
Bosma, W., Cannon, J., Playoust, C.: The magma algebra system. I: the user language. J. Symb. Comput. 24(3–4), 235–265 (1997)
Brooksbank, P.A., Leemans, D.: Polytopes of large rank for \({\rm PSL}(4, {\mathbb{F}}_q)\). J. Algebra 452, 390–400 (2016)
Brooksbank, P.A., Vicinsky, D.A.: Three-dimensional classical groups acting on polytopes. Discrete Comput. Geom. 44(3), 654–659 (2010)
Cameron, P.J., Fernandes, M.E., Leemans, D., Mixer, M.: Highest rank of a polytope for \(A_n\). Proc. Lond. Math. Soc. 115(1), 135–176 (2017)
Conder, M.: The smallest regular polytopes of given rank. Adv. Math. 236, 92–110 (2013)
Conder, M.: Regular polytopes with up to 2000 flags. https://www.math.auckland.ac.nz/~conder/RegularPolytopesWithFewFlags-ByOrder.txt
Conder, M., Cunningham, G.: Tight orientably-regular polytopes. ARS Math. Contemp. 8(1), 69–82 (2015)
Connor, T., De Saedeleer, J., Leemans, D.: Almost simple groups with socle \({\rm PSL}(2, q)\) acting on abstract regular polytopes. J. Algebra 423, 550–558 (2015)
Cunningham, G.: Minimal equivelar polytopes. ARS Math. Contemp. 7(2), 299–315 (2014)
Cunningham, G., Pellicer, D.: Classification of tight regular polyhedra. J. Algebraic Comb. 43(3), 665–691 (2016)
Fernandes, M.E., Leemans, D.: Polytopes of high rank for the symmetric groups. Adv. Math. 228(6), 3207–3222 (2011)
Fernandes, M.E., Leemans, D., Mixer, M.: Polytopes of high rank for the alternating groups. J. Comb. Theory Ser. A 119(1), 42–56 (2012)
Fernandes, M.E., Leemans, D., Mixer, M.: All alternating groups \(A_n\) with \(n\ge 12\) have polytopes of rank \(\bigl \lfloor \frac{n-1}{2}\bigr \rfloor \). SIAM J. Discrete Math. 26(2), 482–498 (2012)
Fernandes, M.E., Leemans, D., Mixer, M.: Corrigendum to “Polytopes of high rank for the symmetric groups”. Adv. Math. 238, 506–508 (2013)
Fernandes, M.E., Leemans, D., Mixer, M.: Extension of the classification of high rank regular polytopes. Trans. Am. Math. Soc. 370(12), 8833–8857 (2018)
Gomi, Y., Loyola, M.L., Peñas, M.L.A.N.: String C-groups of order. Contrib. Discrete Math. 13(1), 1–22 (2018)
Grünbaum, B.: Regularity of graphs, complexes and designs. Problèmes Combinatoires et Théorie des Graphes. In: Colloques Internationaux du Centre National de la Recherche Scientifique, vol. 260, pp. 191–197. CNRS, Paris (1978)
Hou, D.-D., Feng, Y.-Q., Leemans, D.: Existence of regular 3-polytopes of order \(2^n\). J. Group Theory 22(4), 579–616 (2019)
Leemans, D.: Almost simple groups of Suzuki type acting on polytopes. Proc. Am. Math. Soc. 134(12), 3649–3651 (2006)
Leemans, D., Schulte, E.: Groups of type \(L_2(q)\) acting on polytopes. Adv. Geom. 7(4), 529–539 (2007)
Leemans, D., Schulte, E.: Polytopes with groups of type \({\rm PGL}_2(q)\). ARS Math. Contemp. 2(2), 163–171 (2009)
Leemans, D., Vauthier, L.: An atlas of abstract regular polytopes for small groups. Aequ. Math. 72(3), 313–320 (2006)
Loyola, M.L.: String C-groups from groups of order \(2^m\) and exponent at least \(2^{m-3}\) (2016). arXiv:1607.01457v1
McMullen, P., Schulte, E.: Abstract Regular Polytopes. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2002)
Schulte, E., Ivić Weiss, A.: Problems on polytopes, their groups, and realizations. Perlod. Math. Hungar. 53(1–2), 231–255 (2006)
Acknowledgements
This work was supported by the National Natural Science Foundation of China (11571035, 11731002) and the 111 Project of China (B16002). The authors thank two anonymous referees whose comments and suggestions improved this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Editor in Charge: Kenneth Clarkson
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Dedicated to the memory of Branko Grünbaum.
Rights and permissions
About this article
Cite this article
Hou, DD., Feng, YQ. & Leemans, D. On Regular Polytopes of 2-Power Order. Discrete Comput Geom 64, 339–346 (2020). https://doi.org/10.1007/s00454-019-00119-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-019-00119-5