TWISTED GROUPS AND LOCALLY TOROIDAL REGULAR POLYTOPES

In recent years, much work has been done on the classification of abst ract regular polytopes by their local and global topological type. Abs tract 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, th e authors have studied the corresponding classification of abstract re gular polytopes which are globally or locally toroidal. Here, this inv estigation of locally toroidal regular polytopes is continued, with a particular emphasis on polytopes of ranks 5 and 6. For large classes o f such polytopes, their groups are explicitly identified using twistin g operations on quotients of Coxeter groups. In particular, this leads to new classification results which complement those obtained elsewhe re. The method is also applied to describe certain regular polytopes w ith small facets and vertex-figures.