Groups as automorphisms of dessins d'enfants

TL;DR

This paper extends the representation of groups as automorphisms of dessins d'enfants from finite to countable groups using non-compact dessins, and demonstrates that any tame action of such groups can be realized.

Contribution

It provides a constructive proof that all countable groups can be represented as automorphisms of non-compact dessins d'enfants, broadening the scope from finite groups.

Findings

Countable groups can be realized as automorphism groups of non-compact dessins.

Any tame action of a countable group is realizable as a dessin automorphism.

The proof is constructive and straightforward.

Abstract

It is known that every finite group can be represented as the full group of automorphisms of a suitable compact dessin d'enfant. In this paper, we give a constructive and easy proof that the same holds for any countable group by considering non-compact dessins. Moreover, we show that any tame action of a countable group is so realizable.