Maniplexes with automorphism group $\textrm{PSL}_2(q)$

TL;DR

This paper constructs rank 4 highly symmetric maniplexes with automorphism group PSL_2(q) for infinitely many q, and proves no higher rank maniplexes with this automorphism group exist.

Contribution

It demonstrates the existence of infinitely many rank 4 reflexible maniplexes with PSL_2(q) automorphism groups and establishes the non-existence of such maniplexes in higher ranks.

Findings

Existence of rank 4 reflexible maniplexes with PSL_2(q) for infinitely many q.

No reflexible maniplex of rank > 4 has PSL_2(q) as automorphism group.

Abstract

A maniplex of rank $n$ is a combinatorial object that generalises the notion of a rank $n$ abstract polytope. A maniplex with the highest possible degree of symmetry is called reflexible. In this paper we prove that there is a rank $4$ reflexible maniplex with automorphism group $\textrm{PSL}_2(q)$ for infinitely many prime powers $q$, and that no reflexible maniplex of rank $n > 4$ exists that has $\textrm{PSL}_2(q)$ as its full automorphism group.