Twisted Groups and Locally Toroidal Regular Polytopes

McMullen, Peter; Schulte, Egon

doi:10.1090/S0002-9947-96-01561-9

Twisted Groups and Locally Toroidal Regular Polytopes
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by Peter McMullen and Egon Schulte PDF

Trans. Amer. Math. Soc. 348 (1996), 1373-1410 Request permission

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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