Projective linear groups as automorphism groups of chiral polytopes

TL;DR

This paper investigates which projective linear groups can serve as automorphism groups of chiral polytopes, establishing non-existence results for ranks ≥5 and providing conditions for ranks 4, advancing understanding of symmetry groups in geometric structures.

Contribution

It proves that PSL(2,q) and PGL(2,q) are not automorphism groups of chiral polytopes of rank ≥5, and characterizes when PGL(2,q) and PSL(2,q) can be automorphism groups of rank 4 polytopes.

Findings

PSL(2,q) and PGL(2,q) are not automorphism groups of rank ≥5 chiral polytopes.

PGL(2,q) is automorphism group of at least one rank 4 chiral polytope for all q ≥ 5.

Conditions are identified for PSL(2,q) to be automorphism group of rank 4, with some cases remaining open.

Abstract

It is already known that the automorphism group of a chiral polyhedron is never isomorphic to $PSL(2,q)$ or $PGL(2,q)$ for any prime power $q$. In this paper, we show that $PSL(2,q)$ and $PGL(2,q)$ are never automorphism groups of chiral polytopes of rank at least $5$. Moreover, we show that $PGL(2,q)$ is the automorphism group of at least one chiral polytope of rank $4$ for every $q\geq5$. Finally, we determine for which values of $q$ the group $PSL(2,q)$ is the automorphism group of a chiral polytope of rank $4$, except when $q=p^d\equiv3\pmod{4}$ where $d>1$ is not a prime power, in which case the problem remains unsolved.