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Groups Complexity Cryptology

Managing Editor: Rosenberger, Gerhard / Shpilrain, Vladimir

Editorial Board Member: Blackburn, Simon R. / Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / May, Alexander / Mikhalev, Alexander V. / Myasnikov, Alexei / Roman'kov, Vitalii / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura / Weil, Pascal

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Cutting up graphs revisited – a short proof of Stallings' structure theorem

1Faculty of Mathematics, University of Vienna, Norbergstr. 15, 1090 Vienna, Austria. E-mail:

Citation Information: Groups – Complexity – Cryptology. Volume 2, Issue 2, Pages 213–221, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: 10.1515/gcc.2010.013, November 2010

Publication History

Published Online:


This is a short proof of the existence of finite sets of edges in graphs with more than one end, such that after removing them we obtain two components which are nested with all their isomorphic images. This was first done in “Cutting up graphs” [Dunwoody, Combinatorica 2: 15–23, 1982]. Together with a certain tree construction and some elementary Bass–Serre theory this yields a combinatorial proof of Stallings' theorem on the structure of finitely generated groups with more than one end.

Keywords.: Stallings Theorem on groups with more than one end; structure trees

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