An extension of the classification of high rank regular polytopes

Abstract

Up to isomorphism and duality, there are exactly two nondegenerate abstract regular polytopes of rank greater than n−3 (one of rank n−1 and one of rank n−2) with automorphism groups that are transitive permutation groups of degree n ≥ 7. In this paper we extend this classification of high rank regular polytopes to include the ranks n − 3 and n − 4. The result is, up to isomorphism and duality, there are exactly seven abstract regular polytopes of rank n − 3 for each n ≥ 9, and there are nine abstract regular polytopes of rank n−4 for each n ≥ 11. Moreover, we show that if a transitive permutation group Γ of degree n ≥ 11 is the automorphism group of an abstract regularpolytope of rank at least n − 4, then Γ~=S_n.