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A short proof of Greenberg’s Theorem

A short proof of Greenberg’s Theorem
A short proof of Greenberg’s Theorem

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.

Automorphism group, Finitely generated group, Fuchsian group, Greenberg’s Theorem, Non-arithmetic group, Riemann surface, Triangle group
0271-4132
83-88
American Mathematical Society
Jones, Gareth A.
fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5
Wootton, Aaron
Broughton, S. Allen
Paulhus, Jennifer
Jones, Gareth A.
fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5
Wootton, Aaron
Broughton, S. Allen
Paulhus, Jennifer

Jones, Gareth A. (2022) A short proof of Greenberg’s Theorem. In, Wootton, Aaron, Broughton, S. Allen and Paulhus, Jennifer (eds.) Automorphisms of Riemann Surfaces, Subgroups of Mapping Class Groups and Related Topics. (Contemporary Mathematics, 776) American Mathematical Society, pp. 83-88. (doi:10.1090/conm/776/15608).

Record type: Book Section

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.

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More information

Published date: 2022
Additional Information: Publisher Copyright: © 2022 American Mathematical Society. Copyright: Copyright 2022 Elsevier B.V., All rights reserved.
Keywords: Automorphism group, Finitely generated group, Fuchsian group, Greenberg’s Theorem, Non-arithmetic group, Riemann surface, Triangle group

Identifiers

Local EPrints ID: 474845
URI: http://eprints.soton.ac.uk/id/eprint/474845
ISSN: 0271-4132
PURE UUID: d7b36001-c64a-484a-a524-a36843dcd6e2

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Date deposited: 03 Mar 2023 17:46
Last modified: 05 Jun 2024 18:31

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Contributors

Author: Gareth A. Jones
Editor: Aaron Wootton
Editor: S. Allen Broughton
Editor: Jennifer Paulhus

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