Groups of Ree type in characteristic 3 acting on polytopes

TL;DR

This paper explores the automorphism groups of Ree groups in characteristic 3, showing they correspond to regular polyhedra, but larger groups containing them do not form automorphism groups of higher-rank polytopes.

Contribution

It demonstrates that Ree groups in characteristic 3 are automorphism groups of regular polyhedra, while their overgroups are not automorphism groups of any regular polytope.

Findings

Ree groups in characteristic 3 are automorphism groups of regular polyhedra.

Overgroups of Ree groups are not automorphism groups of any regular polytope.

Ree groups in characteristic 3 do not extend to higher-rank regular polytopes.

Abstract

Every Ree group $R(q)$, with $q\neq 3$ an odd power of 3, is the automorphism group of an abstract regular polytope, and any such polytope is necessarily a regular polyhedron (a map on a surface). However, an almost simple group $G$ with $R(q) < G \leq \mathsf{Aut}(R(q))$ is not a C-group and therefore not the automorphism group of an abstract regular polytope of any rank.