Abstract
An \((n+1)\)-toroid is a quotient of a tessellation of the n-dimensional Euclidean space with a lattice group. Toroids are generalisations of maps on the torus to higher dimensions and also provide examples of abstract polytopes. Equivelar toroids are those that are induced by regular tessellations. In this paper we present a classification of equivelar \((n+1)\)-toroids with at most n flag-orbits; in particular, we discuss a classification of 2-orbit toroids of arbitrary dimension.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brehm, U., Kühnel, W.: Equivelar maps on the torus. Eur. J. Comb. 29(8), 1843–1861 (2008)
Conder, M.D.E.: Regular maps and hypermaps of Euler characteristic \(-1\) to \(-200\). J. Comb. Theory Ser. B 99(2), 455–459 (2009)
Conder, M., Dobcsányi, P.: Determination of all regular maps of small genus. J. Comb. Theory Ser. B 81(2), 224–242 (2001)
Coxeter, H.S.M.: Regular Polytopes. Dover, New York (1973)
Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 14. Springer, New York (1972)
Cunningham, G., Del Río-Francos, M., Hubard, I., Toledo, M.: Symmetry type graphs of polytopes and maniplexes. Ann. Comb. 19(2), 243–268 (2015)
Dixon, J.D., Mortimer, B.: Permutation Groups. Graduate Texts in Mathematics, vol. 163. Springer, New York (1996)
Hartley, M.I., McMullen, P., Schulte, E.: Symmetric tessellations on Euclidean space-forms. Can. J. Math. 51(6), 1230–1239 (1999)
Hubard, I.: Two-orbit polyhedra from groups. Eur. J. Comb. 31(3), 943–960 (2010)
Hubard, I., Orbanić, A., Pellicer, D., Ivić Weiss, A.: Symmetries of equivelar 4-toroids. Discrete Comput. Geom. 48(4), 1110–1136 (2012)
Jones, G.A., Singerman, D.: Theory of maps on orientable surfaces. Proc. Lond. Math. Soc. 37(2), 273–307 (1978)
Jones, G., Singerman, D.: Maps, hypermaps and triangle groups. In: The Grothendieck Theory of Dessins d’Enfants (Luminy 1993). London Mathematical Society Lecture Note Series, vol. 200, pp. 115–145. Cambridge University Press, Cambridge (1994)
Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)
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)
Orbanić, A., Pellicer, D., Ivić Weiss, A.: Map operations and \(k\)-orbit maps. J. Comb. Theory Ser. A 117(4), 411–429 (2010)
Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics, vol. 149. Springer, New York (2006)
GAP—Groups, Algorithms, and Programming. Version 4.8.6 (2016). http://www.gap-system.org
Acknowledgements
Both authors where supported by PAPIIT-México under Project Grant IN101615 as well as by the National Council of Science and Technology of Mexico (CONACyT México). The authors also want to thank the anonymous referee as well as Isabel Hubard and Daniel Pellicer for their valuable comments on this work. The research on this paper was developed while the first author was a visiting student in Centro de Ciencias Matemáticas, National Autonomous University of Mexico (UNAM), Unidad Morelia, Mexico, and the second author was a student in the same institution.
Author information
Authors and Affiliations
Corresponding author
Additional information
Editor in Charge: János Pach
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supported by PAPIIT UNAM Project Grant IN101615.
Rights and permissions
About this article
Cite this article
Collins, J., Montero, A. Equivelar Toroids with Few Flag-Orbits. Discrete Comput Geom 65, 305–330 (2021). https://doi.org/10.1007/s00454-020-00230-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-020-00230-y