# A short proof of Greenberg's Theorem

## TL;DR

This paper provides a concise algebraic proof demonstrating that every finitely generated group can be realized as the automorphism group of a compact Riemann surface, extending Greenberg's original theorem.

## Contribution

It offers a shorter, explicit algebraic proof for finitely generated groups, simplifying Greenberg's original argument.

## Key findings

- Finitely generated groups can be realized as automorphism groups of compact Riemann surfaces.
- The proof is shorter and more explicit than previous methods.
- Greenberg's theorem is extended to finitely generated groups with a new proof.

## Abstract

Greenberg proved that every countable group $A$ is isomorphic to the automorphism group of a Riemann surface, which can be taken to be compact if $A$ is finite. We give a short and explicit algebraic proof of this for finitely generated groups $A$.