<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume10 Issue3" %>
Simpler Tests for Semisparse Subgroups
Michael I. Hartley
Faculty of Engineering and Computer Science, University of Nottingham Malaysia Campus, Jalan Broga, Semenyih, 43500 Selangor, Malaysia
Annals of Combinatorics 10 (3) p. 343-352 September, 2006
AMS Subject Classification: 51M20, 52B15, 05E25
The main results of this article facilitate the search for quotients of regular abstract polytopes. A common approach in the study of abstract polytopes is to construct polytopes with specified facets and vertex figures. Any nonregular polytope , Q may be constructed as a quotient of a regular polytope P by a (so-called) semisparse subgroup of its automorphism group W (which will be a string C-group). It becomes important, therefore, to be able to identify whether or not a given subgroup N of a string C-group W is semisparse. This article proves a number of properties of semisparse subgroups. These properties may be used to test for semisparseness in a way which is computationally more efficient than previous methods. The methods are used to find an example of a section regular polytope of type {6, 3, 3} whose facets are Klein bottles.
Keywords: abstract polytope, quotient polytope, semisparse subgroup, klein bottle


1. N. Bourbaki, Groupes et Alg¨¨bres de Lie, Chapitre, IV–VI , Hermann, Paris, 1968.

2. H.S.M. Coxeter, Regular Polytopes, Methuen and Co., Ltd., London, 1948.

3. The GAP Group, GAP - Groups, Algorithms, Programming - A System for Computational Discrete Algebra, Version 4.3, 2002. http://www.gap-system.org.

4. G. Gévay, On perfect 4-polytopes, Beiträge Algebra Geom. 43 (2002) 243–259.

5. M.I. Hartley, Combinatorially regular Euler polytopes, Bull. Austral. Math. Soc. 56 (1997) 173–174.

6. M.I. Hartley, All polytopes are quotients, and isomorphic polytopes are quotients by conjugate subgroups, Discrete Comput. Geom. 21 (1999) 289–298.

7. M.I. Hartley, More on quotient polytopes, Aequationes Math. 57 (1999) 108–120.

8. M.I. Hartley, Polytopes of finite type, Discrete Math. 218 (2000) 97–108.

9. M.I. Hartley, Quotients of some finite universal locally projective polytopes, Discrete Comput. Geom. 29 (2003) 435–443.

10. M.I. Hartley and D. Leemans, Quotients of a universal locally projective polytope of type {5, 3, 5}, Math. Z. 247 (2004) 663–674.

11. J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math. 29, Cambridge University Press, Cambridge, 1990.

12. P. McMullen and E. Schulte, Quotients of polytopes and C-groups, Discrete Comput. Geom. 11 (1994) 453–464.

13. P. McMullen and E. Schulte, Flat regular polytopes, Ann. Combin. 1 (1997) 261–278.

14. P. McMullen and E. Schulte, Abstract Regular Polytopes, Encyclopedia Math. Appl. 92, Cambridge University Press, Cambridge, 2002.

15. E. Schulte, Amalgamation of regular incidence-polytopes, Proc. London Math. Soc. 56 (1988) 303–328.

16. E. Schulte, Classification of locally toroidal regular polytopes, In: Polytopes: Abstract, Convex and Computational, T. Bisztriczky et al., Eds., Kluwer Acad. Publ., (1994) pp. 125–154.

17. A.I. Weiss, Incidence polytopes of type {6, 3, 3}, Geom. Dedicata 20 (1986) 147–155.