There is an increasingly extensive literature on the problem of describing the connection (monodromy) groups and automorphism groups of families of polytopes and maniplexes that are not regular or reflexible. Many such polytopes and maniplexes arise as the result of constructions such as truncations and products. Here we show that for a wide variety of these constructions, the connection group of the output can be described in a nice way in terms of the connection group of the input. We call such operations stratified. Moreover, we show that, if $F$ is a maniplex operation in one of two broad subclasses of stratified operations, and if $\mathcal{R}$ is the smallest reflexible cover of some maniplex $\mathcal{M}$, then the connection group of $F\left(\mathcal{R}\right)$ is equal to the connection group of $F\left(\mathcal{M}\right)$. In particular, we show that this is true for truncations and medials of maps, for products of polytopes (including pyramids and prisms over polytopes), and for the mix of maniplexes. As an application, we determine the smallest reflexible covers of the pyramids over the equivelar toroidal maps.

Revised:

Accepted:

Published online:

Keywords: Polytope, maniplex, connection group, monodromy group, truncation, medial, pyramid, prism

^{1}; Pellicer, Daniel

^{2}; Williams, Gordon

^{3}

@article{ALCO_2022__5_2_267_0, author = {Cunningham, Gabe and Pellicer, Daniel and Williams, Gordon}, title = {Stratified operations on maniplexes}, journal = {Algebraic Combinatorics}, pages = {267--287}, publisher = {The Combinatorics Consortium}, volume = {5}, number = {2}, year = {2022}, doi = {10.5802/alco.208}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.208/} }

TY - JOUR AU - Cunningham, Gabe AU - Pellicer, Daniel AU - Williams, Gordon TI - Stratified operations on maniplexes JO - Algebraic Combinatorics PY - 2022 SP - 267 EP - 287 VL - 5 IS - 2 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.208/ DO - 10.5802/alco.208 LA - en ID - ALCO_2022__5_2_267_0 ER -

%0 Journal Article %A Cunningham, Gabe %A Pellicer, Daniel %A Williams, Gordon %T Stratified operations on maniplexes %J Algebraic Combinatorics %D 2022 %P 267-287 %V 5 %N 2 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.208/ %R 10.5802/alco.208 %G en %F ALCO_2022__5_2_267_0

Cunningham, Gabe; Pellicer, Daniel; Williams, Gordon. Stratified operations on maniplexes. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 267-287. doi : 10.5802/alco.208. https://alco.centre-mersenne.org/articles/10.5802/alco.208/

[1] The monodromy group of the $n$-pyramid, Discrete Math., Volume 320 (2014), pp. 55-63 | DOI | MR | Zbl

[2] The monodromy group of a truncated simplex, J. Algebraic Combin., Volume 42 (2015) no. 3, pp. 745-761 | DOI | MR | Zbl

[3] Equivelar maps on the torus, European J. Combin., Volume 29 (2008) no. 8, pp. 1843-1861 | DOI | MR | Zbl

[4] The smallest regular polytopes of given rank, Adv. Math., Volume 236 (2013), pp. 92-110 | DOI | MR | Zbl

[5] Self-dual, self-Petrie covers of regular polyhedra, Symmetry, Volume 4 (2012) no. 1, pp. 208-218 | DOI | MR | Zbl

[6] Flat extensions of abstract polytopes, Art Discrete Appl. Math., Volume 4 (2021) no. 3, Paper no. 3.06, 14 pages | DOI | MR

[7] Symmetry type graphs of polytopes and maniplexes, Ann. Comb., Volume 19 (2015) no. 2, pp. 243-268 | DOI | MR | Zbl

[8] Open problems on $k$-orbit polytopes, Discrete Math., Volume 341 (2018) no. 6, pp. 1645-1661 | DOI | MR | Zbl

[9] Regular incidence-complexes and dimensionally unbounded sequences of such. I, Convexity and graph theory (Jerusalem, 1981) (North-Holland Math. Stud.), Volume 87, North-Holland, Amsterdam, 1984, pp. 115-127 | DOI | MR | Zbl

[10] The twist operator on maniplexes, Discrete geometry and symmetry (Springer Proc. Math. Stat.), Volume 234, Springer, Cham, 2018, pp. 127-145 | DOI | MR | Zbl

[11] Minimal covers of equivelar toroidal maps, Ars Math. Contemp., Volume 9 (2015) no. 1, pp. 77-91 | DOI | MR | Zbl

[12] Polytopes of high rank for the alternating groups, J. Combin. Theory Ser. A, Volume 119 (2012) no. 1, pp. 42-56 | DOI | MR | Zbl

[13] GAP – Groups, Algorithms, and Programming, Version 4.4.12 (2008) (http://www.gap-system.org)

[14] Polytopality of maniplexes, Discrete Math., Volume 341 (2018) no. 7, pp. 2068-2079 | DOI | MR | Zbl

[15] Products of abstract polytopes, J. Combin. Theory Ser. A, Volume 157 (2018), pp. 287-320 | DOI | MR | Zbl

[16] Convex polytopes, Graduate Texts in Mathematics, 221, Springer-Verlag, New York, 2003, xvi+468 pages (Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler) | DOI | MR | Zbl

[17] Polytopes derived from sporadic simple groups, Contrib. Discrete Math., Volume 5 (2010) no. 2, pp. 106-118 | MR | Zbl

[18] Minimal covers of the prisms and antiprisms, Discrete Math., Volume 312 (2012) no. 20, pp. 3046-3058 | DOI | MR | Zbl

[19] Representing the sporadic Archimedean polyhedra as abstract polytopes, Discrete Math., Volume 310 (2010) no. 12, pp. 1835-1844 | DOI | MR | Zbl

[20] Constructions of k-orbit Abstract Polytopes, ProQuest LLC, Ann Arbor, MI, 2013, 147 pages Thesis (Ph.D.)–Northeastern University | MR

[21] Two-orbit polyhedra from groups, European J. Combin., Volume 31 (2010) no. 3, pp. 943-960 | DOI | MR | Zbl

[22] Medial symmetry type graphs, Electron. J. Combin., Volume 20 (2013) no. 3, Paper no. 29, 28 pages | DOI | MR | Zbl

[23] Symmetries of equivelar 4-toroids, Discrete Comput. Geom., Volume 48 (2012) no. 4, pp. 1110-1136 | DOI | MR | Zbl

[24] From geometry to groups and back: The study of highly symmetric polytopes, ProQuest LLC, Ann Arbor, MI, 2007, 168 pages Thesis (Ph.D.)–York University (Canada) | MR

[25] Maps admitting trialities but not dualities, European J. Combin., Volume 31 (2010) no. 7, pp. 1805-1818 | DOI | MR | Zbl

[26] Flag bicolorings, pseudo-orientations, and double covers of maps, Electron. J. Combin., Volume 24 (2017) no. 1, Paper no. 1.3, 23 pages | DOI | MR | Zbl

[27] Abstract regular polytopes, Encyclopedia of Mathematics and its Applications, 92, Cambridge University Press, Cambridge, 2002, xiv+551 pages | DOI | MR | Zbl

[28] Minimal covers of the Archimedean tilings, part II, Electron. J. Combin., Volume 20 (2013) no. 2, Paper no. 20, 19 pages | DOI | MR | Zbl

[29] Mixing and monodromy of abstract polytopes, Trans. Amer. Math. Soc., Volume 366 (2014) no. 5, pp. 2651-2681 | DOI | MR | Zbl

[30] $F$-actions and parallel-product decomposition of reflexible maps, J. Algebraic Combin., Volume 26 (2007) no. 4, pp. 507-527 | DOI | MR | Zbl

[31] Map operations and $k$-orbit maps, J. Combin. Theory Ser. A, Volume 117 (2010) no. 4, pp. 411-429 | DOI | MR | Zbl

[32] Cleaved abstract polytopes, Combinatorica, Volume 38 (2018) no. 3, pp. 709-737 | DOI | MR | Zbl

[33] Minimal covers of the Archimedean tilings, Part 1, Electron. J. Combin., Volume 19 (2012) no. 3, Paper no. 6, 37 pages | MR | Zbl

[34] Pyramids over regular 3-tori, SIAM J. Discrete Math., Volume 32 (2018) no. 1, pp. 249-265 | DOI | MR | Zbl

[35] Self-dual and self-Petrie-dual regular maps, J. Graph Theory, Volume 69 (2012) no. 2, pp. 152-159 | DOI | MR | Zbl

[36] Amalgamation of regular incidence-polytopes, Proc. London Math. Soc. (3), Volume 56 (1988) no. 2, pp. 303-328 | DOI | MR | Zbl

[37] Regular combinatorial maps, J. Combin. Theory Ser. B, Volume 35 (1983) no. 3, pp. 256-277 | DOI | MR | Zbl

[38] Parallel products in groups and maps, J. Algebra, Volume 167 (1994) no. 3, pp. 539-546 | DOI | MR | Zbl

[39] Maniplexes: Part 1: maps, polytopes, symmetry and operators, Symmetry, Volume 4 (2012) no. 2, pp. 265-275 | DOI | MR | Zbl

*Cited by Sources: *