Invariant Meromorphic Functions on the Wallpaper Groups

TL;DR

This paper constructs meromorphic functions invariant under wallpaper group symmetries, addressing a question about elliptic function symmetries and exploring potential generalizations to all wallpaper groups.

Contribution

It provides explicit constructions of invariant meromorphic functions for wallpaper groups and discusses their possible extensions to all wallpaper groups.

Findings

Explicit invariant meromorphic functions for certain wallpaper groups

Addresses a question on elliptic function symmetries

Explores potential generalizations to all wallpaper groups

Abstract

We construct the meromorphic functions invariant under the action of the sense-preserving wallpaper groups on the complex plane. We discuss possible generalisa-tions of this to the general wallpaper groups. This provides the answer to a question posed on StackExchange regarding the possible symmetries of elliptic functions.