Semiregular abstract polyhedra with trivial facet stabilizer

TL;DR

This paper constructs semiregular abstract polyhedra with trivial facet stabilizer, demonstrating that such polyhedra can have many flag orbits and few facet orbits, and explores their construction and generalization.

Contribution

It introduces a method to construct semiregular, facet-transitive polyhedra with trivial facet stabilizer and multiple flag orbits, expanding understanding of abstract polyhedra symmetry.

Findings

Existence of semiregular polyhedra with unbounded flag orbits

Construction of alternating semiregular polyhedra with two facet orbits

Potential for generalizing constructions to higher ranks

Abstract

Abstract polytopes generalize the face lattice of convex polytopes. A polytope is semiregular if its facets are regular and its automorphism group acts transitively on its vertices. In this paper we construct semiregular, facet-transitive polyhedra with trivial facet stabilizer, showing that semiregular abstract polyhedra can have an unbounded number of flag orbits, while having as little as one facet orbit. We interpret this construction in terms of operations applied to high rank regular and chiral polytopes, and we see how this same operations help us construct alternating semiregular polyhedra (that is, with two facet orbits and adjacent facets in different orbits). Finally, we give an idea to generalize this construction giving examples in higher ranks.