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Arc-transitive maps with coprime Euler characteristic and edge number -- II

Cai Heng Li, Luyi Liu

TL;DR

This work completes the classification of arc-transitive maps whose Euler characteristic ฯ‡(๐“œ) is coprime to the edge count |E|, focusing on non-solvable automorphism groups. It shows that either the map is flag-regular with G โ‰… A5 on the complete graph K6 or Petersen, or the map is a reversing map RevMap(G,x,y,z) with G โ‰… PSL(2,p) or (Z_m ร— PSL(2,p)) : Z2 and explicit dihedral stabilizers {D_{2p},D_{p+1},D_{p-1}} (or their 2-fold analogues); it further provides constructive methods for reversing triples via PSL/PGL actions and analyzes stabilisers to classify all such reversing maps. The paper thus advances the symmetry-based taxonomy of arc-transitive maps and introduces new reversing-map families linked to finite linear groups.

Abstract

This is the second of a series of papers which aim towards a classification of edge-transitive maps of which the Euler characteristic and the edge number are coprime. This one carries out the classification work for arc-transitive maps with non-solvable automorphism groups, which together with the first one completes a description of arc-transitive maps with the Euler characteristic and the edge number coprime. The classification is involved with a construction of some new and interesting reversing maps.

Arc-transitive maps with coprime Euler characteristic and edge number -- II

TL;DR

This work completes the classification of arc-transitive maps whose Euler characteristic ฯ‡(๐“œ) is coprime to the edge count |E|, focusing on non-solvable automorphism groups. It shows that either the map is flag-regular with G โ‰… A5 on the complete graph K6 or Petersen, or the map is a reversing map RevMap(G,x,y,z) with G โ‰… PSL(2,p) or (Z_m ร— PSL(2,p)) : Z2 and explicit dihedral stabilizers {D_{2p},D_{p+1},D_{p-1}} (or their 2-fold analogues); it further provides constructive methods for reversing triples via PSL/PGL actions and analyzes stabilisers to classify all such reversing maps. The paper thus advances the symmetry-based taxonomy of arc-transitive maps and introduces new reversing-map families linked to finite linear groups.

Abstract

This is the second of a series of papers which aim towards a classification of edge-transitive maps of which the Euler characteristic and the edge number are coprime. This one carries out the classification work for arc-transitive maps with non-solvable automorphism groups, which together with the first one completes a description of arc-transitive maps with the Euler characteristic and the edge number coprime. The classification is involved with a construction of some new and interesting reversing maps.
Paper Structure (7 sections, 14 theorems, 34 equations, 1 table)

This paper contains 7 sections, 14 theorems, 34 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A reduction to reversing maps
  4. The proof of Theorem \ref{['mainThm']}
  5. Stabilizers
  6. Arc-regular triples
  7. Completing the proof of Theorem \ref{['mainThm']}

Key Result

Theorem 1.1

Let ${\mathcal{M}}=(V,E,F)$ be a map, and let $G\leqslant\mathrm{Aut}({\mathcal{M}})$ be arc-transitive on ${\mathcal{M}}$. If $\gcd(\chi({\mathcal{M}}),|E|)=1$ and $G$ is non-solvable, then $G=\mathrm{Aut}({\mathcal{M}})$, ${\mathcal{M}}$ is non-orientable, and further either

Theorems & Definitions (23)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2: LiuLY-1 Lemma 3.2
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 13 more