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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Twisted Groups and
Locally Toroidal Regular Polytopes

Authors: Peter McMullen and Egon Schulte
Journal: Trans. Amer. Math. Soc. 348 (1996), 1373-1410
MSC (1991): Primary 51M20
MathSciNet review: 1340181
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Abstract: In recent years, much work has been done on the classification of abstract regular polytopes by their local and global topological type. Abstract regular polytopes are combinatorial structures which generalize the well-known classical geometric regular polytopes and tessellations. In this context, the classical theory is concerned with those which are of globally or locally spherical type. In a sequence of papers, the authors have studied the corresponding classification of abstract regular polytopes which are globally or locally toroidal. Here, this investigation of locally toroidal regular polytopes is continued, with a particular emphasis on polytopes of ranks $5$ and $6$. For large classes of such polytopes, their groups are explicitly identified using twisting operations on quotients of Coxeter groups. In particular, this leads to new classification results which complement those obtained elsewhere. The method is also applied to describe certain regular polytopes with small facets and vertex-figures.

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Additional Information

Peter McMullen
Affiliation: Department of Mathematics, University College London, Gower Street, London WCIE 6BT, England

Egon Schulte
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115

Keywords: Polyhedra and polytopes; regular figures, division of space
Received by editor(s): January 7, 1995
Additional Notes: Supported by NSF grant DMS-9202071
Article copyright: © Copyright 1996 American Mathematical Society