Simpler Tests for Semisparse Subgroups

Abstract.

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 $\mathcal{Q}$ may be constructed as a quotient of a regular polytope $\mathcal{P}$ 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.