Highly symmetric unstable maniplexes

TL;DR

This paper investigates the symmetry properties of maniplexes, a generalization of maps and polytopes, demonstrating the existence of highly symmetric unstable maniplexes with two flag-orbits for all ranks greater than two.

Contribution

It proves the existence of 2-orbit unstable maniplexes for every rank greater than two, expanding understanding of symmetry in maniplexes beyond known cases.

Findings

Regular maniplexes are always stable.

Unstable maniplexes with 4 flag-orbits are known for rank 3.

Existence of 2-orbit maniplexes for all ranks n > 2.

Abstract

A maniplex of rank n s an n-valent properly edge-coloured graph that generalises, simultaneously, maps on surfaces and abstract polytopes. The problem of stability in maniplexes is a natural variant of the problem of stability in graphs. A maniplex is stable if every automorphism of its canonical double cover is a lift of some automorphism of the original maniplex. Due to their very rich structure, regular (maximally symmetric) maniplexes are always stable. It is thus natural to ask what is the maximum possible degree of symmetry that a maniplex that is not stable can admit. Symmetry in maniplexes is usually measured by the number of orbits on flags (nodes) of their automorphism group. A few families of unstable maniplexes with 4 flag-orbits are known for rank 3. In this paper, we show that 2-orbit maniplexes exist for every rank n > 2$.