Automorphism groups of maps, hypermaps and dessins

TL;DR

This paper provides a detailed proof of a theorem characterizing the automorphism groups of maps, hypermaps, dessins, and related structures, with applications to their symmetries and classifications.

Contribution

It offers a comprehensive proof of a theorem on the centraliser of transitive permutation groups and explores automorphism groups in various categories, including primitive and non-connected cases.

Findings

Automorphism groups are described via centralisers of monodromy groups.

Counterexamples show limitations of the theorem for infinite objects.

Automorphism groups of primitive monodromy objects are characterized.

Abstract

A detailed proof is given of a theorem describing the centraliser of a transitive permutation group, with applications to automorphism groups of objects in various categories of maps, hypermaps, dessins, polytopes and covering spaces, where the automorphism group of an object is the centraliser of its monodromy group. An alternative form of the theorem, valid for finite objects, is discussed, with counterexamples based on Baumslag--Solitar groups to show how it fails more generally. The automorphism groups of objects with primitive monodromy groups are described, as are those of non-connected objects.