An extension of the classification of high rank regular polytopes

TL;DR

This paper extends the classification of high-rank regular polytopes, identifying new families of such polytopes of ranks n-3 and n-4 for large n, and characterizes their automorphism groups.

Contribution

It expands the known classification of regular polytopes to include ranks n-3 and n-4, providing explicit counts and group isomorphism results for large n.

Findings

Seven regular polytopes of rank n-3 for n≥9

Nine regular polytopes of rank n-4 for n≥11

Automorphism groups are isomorphic to S_n for large n

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 $n\geq 7$. 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 $n\geq 9$, and nine abstract regular polytopes of rank $n-4$ for each $n \geq 11$. Moreover we show that if a transitive permutation group $\Gamma$ of degree $n \geq 11$ is the automorphism group of an abstract regular polytope of rank at least $n-4$, then $\Gamma\cong S_n$.