An extension of the classification of high rank regular polytopes

Abstract

Up to isomorphism and duality, there are exactly two non-degenerate 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 $n\ge 7$. 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 a isomorphism and duality, seven abstract regular polytopes of rank $n-3$ for each $n\ge 9$, and nine abstract regular polytopes of rank $n-4$ for each $n\ge 11$. Moreover we show that if a transitive permutation group $\Gamma$ of degree $n\ge 11$ is the automorphism group of an abstract regular polytope of rank at least $n-4$, then $\Gamma \cong S_n$.