Realisation of groups as automorphism groups in categories

TL;DR

This paper demonstrates that in various mathematical categories, any countable group can be realized as the automorphism group of uncountably many objects, including finite ones, using constructions involving hypermaps and triangle groups.

Contribution

It provides a general method to realize arbitrary groups as automorphism groups in categories of maps, hypermaps, and dessins d'enfants, extending previous results.

Findings

Any countable group is automorphism group of uncountably many objects.

Finite groups can be realized as automorphism groups of finite objects.

Constructs a regular map containing all finite groups as automorphism groups.

Abstract

It is shown that in various categories, including many consisting of maps or hypermaps, oriented or unoriented, of a given hyperbolic type, every countable group $A$ is isomorphic to the automorphism group of uncountably many non-isomorphic objects, infinitely many of them finite if $A$ is finite. In particular, this applies to dessins d'enfants, regarded as finite oriented hypermaps. The proof, involving maximal subgroups of various triangle groups, yields a simple construction of a regular map whose automorphism group contains an isomorphic copy of every finite group.