On the automorphism groups of certain branched structures on surfaces

TL;DR

This paper proves that any finite group can be realized as the automorphism group of a translation surface with poles, and extends these results to branched projective structures, highlighting the diversity of symmetries possible.

Contribution

It demonstrates that all finite groups can be realized as automorphism groups of translation surfaces with poles and extends these findings to branched projective structures.

Findings

Any finite group appears as an automorphism group of some translation surface with poles.

Existence of structures with maximal automorphism groups for given genus.

Extension of results to branched projective structures.

Abstract

We consider translation surfaces with poles on surfaces. We shall prove that any finite group appears as the automorphism group of some translation surface with poles. As a direct consequence we obtain the existence of structures achieving the maximal possible number of automorphisms allowed by their genus and we finally extend the same results to branched projective structures.