Automorphism groups of maps, hypermaps and dessins

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 any 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 can fail in the infinite case. The automorphism groups of objects with primitive monodromy groups are described, as are those of non-connected objects.