Covers ℘ for Abstract Regular Polytopes $\mathcal {Q}$ such that $\mathcal{Q}=\mathcal{P}/\mathbf{Z}_{p}^{k}$

Abstract

One key problem in the theory of abstract polytopes is the so-called amalgamation problem. In its most general form, this is the problem of characterising the polytopes with given facets $\mathcal {K}$ and vertex figures ℒ. The first step in solving it for particular $\mathcal{K}$ and ℒ is to find the universal such polytope, which covers all the others. This article explains a construction that may be attempted on an arbitrary polytope ℘, which often yields an infinite family of finite polytopes covering ℘ and sharing its facets and vertex figures. The existence of such an infinite family proves that the universal polytope is infinite; alternatively, the construction can produce an explicit example of an infinite polytope of the desired type. An algorithm for attempting the construction is explained, along with sufficient conditions for it to work. The construction is applied to a few $\mathcal{K}$ and ℒ for which it was previously not known whether or not the universal polytope was infinite, or for which only a finite number of finite polytopes was previously known. It is conjectured that the construction is quite broadly applicable.