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Highly symmetric unstable maniplexes

Isabel Hubard, Micael Toledo

TL;DR

This work addresses the problem of stability in maniplexes by constructing highly symmetric unstable examples. It introduces cross-cover and colour-coded extension techniques to produce unstable, fully transitive 2-orbit maniplexes of any rank $n>2$, starting from non-orientable regular maps and propagating symmetry via Aut-invariant colourings. The key advance is a two-step construction: seed a proper pair $( m M,oldsymbol heta)$ to obtain an unstable 2-orbit cross-cover ${ m M}^oldsymbol heta$, then extend rank by colourings to yield $2^{n-3}$ non-isomorphic unstable $n$-maniplexes with prescribed 3-face structure; variations of colourings yield distinct symmetry-types. The results significantly expand the catalog of unstable maniplexes and provide a scalable method to produce infinite families with strong transitivity properties, enriching the interplay between stability, symmetry, and higher-rankcombinatorial structures.

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$.

Highly symmetric unstable maniplexes

TL;DR

This work addresses the problem of stability in maniplexes by constructing highly symmetric unstable examples. It introduces cross-cover and colour-coded extension techniques to produce unstable, fully transitive 2-orbit maniplexes of any rank , starting from non-orientable regular maps and propagating symmetry via Aut-invariant colourings. The key advance is a two-step construction: seed a proper pair to obtain an unstable 2-orbit cross-cover , then extend rank by colourings to yield non-isomorphic unstable -maniplexes with prescribed 3-face structure; variations of colourings yield distinct symmetry-types. The results significantly expand the catalog of unstable maniplexes and provide a scalable method to produce infinite families with strong transitivity properties, enriching the interplay between stability, symmetry, and higher-rankcombinatorial structures.

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$.
Paper Structure (11 sections, 21 theorems, 25 equations, 2 figures)

This paper contains 11 sections, 21 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Maniplexes
  3. Stability
  4. On maps
  5. Cross covers
  6. Connectivity
  7. Automorphisms
  8. Cross-covers of maniplexes
  9. Extensions
  10. Extended weight functions
  11. Construction

Key Result

Theorem 1.1

For every non-orientable regular map ${\mathcal{M}}$ of type $\{p,q\}$ where at least one of $p$ or $q$ is odd, there exists an unstable fully transitive $2$-orbit map ${\mathcal{M}}^\omega$, that covers ${\mathcal{M}}$. Moreover, for all $n>3$ there exist $2^{n-3}$ non-isomorphic unstable $2$-orbit

Figures (2)

  • Figure 2.1: The seven possible symmetry-type graphs for a $2$-orbit $3$-maniplex
  • Figure 6.1: The symmetry-type graphs $2^{n}_{\{\overline{0,2}\}}$ and $2^{n}_{\{\overline{1}\}}$

Theorems & Definitions (42)

  • Theorem 1.1
  • Definition 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 32 more