English

A short proof of Greenberg's Theorem

Group Theory 2019-12-17 v2

Abstract

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

Keywords

Cite

@article{arxiv.1908.06675,
  title  = {A short proof of Greenberg's Theorem},
  author = {Gareth A. Jones},
  journal= {arXiv preprint arXiv:1908.06675},
  year   = {2019}
}

Comments

5 pages. This extends the proof in version 1 for finite groups $A$ to all finitely generated groups $A$. Some citations have been added

R2 v1 2026-06-23T10:50:41.930Z