Abelian covers of chiral polytopes

Abstract

Abstract polytopes are combinatorial structures with certain properties drawn from the study of geometric structures, like the Platonic solids, and of maps on surfaces. Of particular interest are the polytopes with maximal possible symmetry (subject to certain natural constraints). Symmetry can be measured by the effect of automorphisms on the ‘flags’ of the polytope, which are maximal chains of elements of increasing rank (dimension). An abstract polytope of rank n is said to be chiral if its automorphism group has precisely two orbits on the flags, such that two flags that differ in one element always lie in different orbits. Examples of chiral polytopes have been difficult to find and construct. In this paper, we introduce a new covering method that allows the construction of some infinite families of chiral polytopes, with each member of a family having the same rank as the original, but with the size of the members of the family growing linearly with one (or more) of the parameters making up its ‘type’ (Schläfli symbol). In particular, we use this method to construct several new infinite families of chiral polytopes of ranks 3, 4, 5 and 6.