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Mixing and monodromy of abstract polytopes


Authors: B. Monson, Daniel Pellicer and Gordon Williams
Journal: Trans. Amer. Math. Soc. 366 (2014), 2651-2681
MSC (2010): Primary 51M20; Secondary 20F55
DOI: https://doi.org/10.1090/S0002-9947-2013-05954-5
Published electronically: November 4, 2013
MathSciNet review: 3165650
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Abstract | References | Similar Articles | Additional Information

Abstract: The monodromy group $ \mathrm {Mon}(\mathcal {P})$ of an $ n$-polytope $ \mathcal {P}$ encodes the combinatorial information needed to construct $ \mathcal {P}$. By applying tools such as mixing, a natural group-theoretic operation, we develop various criteria for $ \mathrm {Mon}(\mathcal {P})$ itself to be the automorphism group of a regular $ n$-polytope $ \mathcal {R}$. We examine what this can say about regular covers of $ \mathcal {P}$, study a peculiar example of a $ 4$-polytope with infinitely many distinct, minimal regular covers, and then conclude with a brief application of our methods to chiral polytopes.


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  • [1] Antonio Breda D'Azevedo, Gareth Jones, Roman Nedela, and Martin Škoviera, Chirality groups of maps and hypermaps, J. Algebraic Combin. 29 (2009), no. 3, 337-355. MR 2496311 (2010g:05159), https://doi.org/10.1007/s10801-008-0138-z
  • [2] Antonio Breda D'Azevedo, Gareth A. Jones, and Egon Schulte, Constructions of chiral polytopes of small rank, Canad. J. Math. 63 (2011), no. 6, 1254-1283. MR 2894438, https://doi.org/10.4153/CJM-2011-033-4
  • [3] Robin P. Bryant and David Singerman, Foundations of the theory of maps on surfaces with boundary, Quart. J. Math. Oxford Ser. (2) 36 (1985), no. 141, 17-41. MR 780347 (86f:57008), https://doi.org/10.1093/qmath/36.1.17
  • [4] Marston Conder, Isabel Hubard, and Tomaž Pisanski, Constructions for chiral polytopes, J. Lond. Math. Soc. (2) 77 (2008), no. 1, 115-129. MR 2389920 (2009b:52031), https://doi.org/10.1112/jlms/jdm093
  • [5] G. Cunningham, Internal and External Invariance of Abstract Polytopes, Ph.D. Thesis, Northeastern University, 2012.
  • [6] Gabe Cunningham, Mixing regular convex polytopes, Discrete Math. 312 (2012), no. 4, 763-771. MR 2872918, https://doi.org/10.1016/j.disc.2011.11.014
  • [7] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.3 (2002), +(http://www.gap-system.org)+.
  • [8] M. I. Hartley, All polytopes are quotients, and isomorphic polytopes are quotients by conjugate subgroups, Discrete Comput. Geom. 21 (1999), no. 2, 289-298. MR 1668118 (2000a:20091), https://doi.org/10.1007/PL00009422
  • [9] Michael I. Hartley, More on quotient polytopes, Aequationes Math. 57 (1999), no. 1, 108-120. MR 1675753 (2000g:51016), https://doi.org/10.1007/s000100050073
  • [10] Michael I. Hartley, Simpler tests for semisparse subgroups, Ann. Comb. 10 (2006), no. 3, 343-352. MR 2284275 (2007k:51025), https://doi.org/10.1007/s00026-006-0292-8
  • [11] M. I. Hartley, The Atlas of Small Regular Polytopes,
    (+http://www.abstract-polytopes.com/atlas+)
  • [12] Michael I. Hartley and Gordon I. Williams, Representing the sporadic Archimedean polyhedra as abstract polytopes, Discrete Math. 310 (2010), no. 12, 1835-1844. MR 2610288 (2011g:52023), https://doi.org/10.1016/j.disc.2010.01.012
  • [13] M. I. Hartley, I. Hubard and D. Leemans, An Atlas of Chiral Polytopes for Small Almost Simple Groups, (+http://www.math.auckland.ac.nz/ dleemans/CHIRAL/+)
  • [14] Isabel Hubard, Alen Orbanić, and Asia Ivić Weiss, Monodromy groups and self-invariance, Canad. J. Math. 61 (2009), no. 6, 1300-1324. MR 2588424 (2011a:52032), https://doi.org/10.4153/CJM-2009-061-5
  • [15] G. A. Jones and J. S. Thornton, Operations on maps, and outer automorphisms, J. Combin. Theory Ser. B 35 (1983), no. 2, 93-103. MR 733017 (85m:05036), https://doi.org/10.1016/0095-8956(83)90065-5
  • [16] Gareth Jones and David Singerman, Maps, hypermaps and triangle groups, The Grothendieck theory of dessins d'enfants (Luminy, 1993) London Math. Soc. Lecture Note Ser., vol. 200, Cambridge Univ. Press, Cambridge, 1994, pp. 115-145. MR 1305395 (95m:20055)
  • [17] Peter McMullen and Egon Schulte, Abstract regular polytopes, Encyclopedia of Mathematics and its Applications, vol. 92, Cambridge University Press, Cambridge, 2002. MR 1965665 (2004a:52020)
  • [18] Barry Monson, Daniel Pellicer, and Gordon Williams, The tomotope, Ars Math. Contemp. 5 (2012), no. 2, 355-370. MR 2945978
  • [19] B. Monson and Egon Schulte, Reflection groups and polytopes over finite fields. II, Adv. in Appl. Math. 38 (2007), no. 3, 327-356. MR 2301701 (2008b:20045), https://doi.org/10.1016/j.aam.2005.12.001
  • [20] B. Monson and Egon Schulte, Semiregular polytopes and amalgamated C-groups, Adv. Math. 229 (2012), no. 5, 2767-2791. MR 2889145, https://doi.org/10.1016/j.aim.2011.12.027
  • [21] B. Monson and E. Schulte, Finite Polytopes have Finite Regular Covers, in preparation.
  • [22] Alen Orbanić, $ F$-actions and parallel-product decomposition of reflexible maps, J. Algebraic Combin. 26 (2007), no. 4, 507-527. MR 2341863 (2008m:05145), https://doi.org/10.1007/s10801-007-0069-0
  • [23] Alen Orbanić, Daniel Pellicer, and Asia Ivić Weiss, Map operations and $ k$-orbit maps, J. Combin. Theory Ser. A 117 (2010), no. 4, 411-429. MR 2592891 (2011a:51015), https://doi.org/10.1016/j.jcta.2009.09.001
  • [24] Daniel Pellicer, A construction of higher rank chiral polytopes, Discrete Math. 310 (2010), no. 6-7, 1222-1237. MR 2579855 (2011b:52012), https://doi.org/10.1016/j.disc.2009.11.034
  • [25] Daniel Pellicer, Developments and open problems on chiral polytopes, Ars Math. Contemp. 5 (2012), no. 2, 333-354. MR 2929596
  • [26] Egon Schulte and Asia Ivić Weiss, Chiral polytopes, Applied geometry and discrete mathematics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, Amer. Math. Soc., Providence, RI, 1991, pp. 493-516. MR 1116373 (92f:51018)
  • [27] Egon Schulte and Asia Ivić Weiss, Chirality and projective linear groups, Discrete Math. 131 (1994), no. 1-3, 221-261. MR 1287736 (95g:52016), https://doi.org/10.1016/0012-365X(94)90387-5
  • [28] Andrew Vince, Combinatorial maps, J. Combin. Theory Ser. B 34 (1983), no. 1, 1-21. MR 701167 (84i:05048), https://doi.org/10.1016/0095-8956(83)90002-3
  • [29] Andrew Vince, Regular combinatorial maps, J. Combin. Theory Ser. B 35 (1983), no. 3, 256-277. MR 735194 (85i:05129), https://doi.org/10.1016/0095-8956(83)90053-9
  • [30] Stephen E. Wilson, Parallel products in groups and maps, J. Algebra 167 (1994), no. 3, 539-546. MR 1287058 (95d:20067), https://doi.org/10.1006/jabr.1994.1200
  • [31] Steve Wilson, Uniform maps on the Klein bottle, J. Geom. Graph. 10 (2006), no. 2, 161-171. MR 2324592 (2008b:51011)
  • [32] Steve Wilson, Maniplexes: Part 1: maps, polytopes, symmetry and operators, Symmetry 4 (2012), no. 2, 265-275. MR 2949129, https://doi.org/10.3390/sym4020265

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Additional Information

B. Monson
Affiliation: Department of Mathematics, University of New Brunswick, Fredericton, New Brunswick, Canada
Email: bmonson@unb.ca

Daniel Pellicer
Affiliation: Instituto de Matematicas, UNAM Morelia, Morelia, Michoacán, México
Email: pellicer@matmor.unam.mx

Gordon Williams
Affiliation: Department of Mathematics, University of Alaska Fairbanks, Fairbanks, Alaska 99775
Email: giwilliams@alaska.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-05954-5
Received by editor(s): June 19, 2012
Received by editor(s) in revised form: September 6, 2012
Published electronically: November 4, 2013
Additional Notes: The first author was supported by NSERC of Canada Discovery Grant # 4818.
The second author was supported by grants PAPIIT (IAOCD) # IA101311 and PAPIIT #IB100312.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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