Groups Complexity Cryptology
Managing Editor: Shpilrain, Vladimir / Weil, Pascal
Editorial Board Member: Blackburn, Simon R. / Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Schäge, Sven / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura
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Random van Kampen diagrams and algorithmic problems in groups
1Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030, USA.
Citation Information: Groups – Complexity – Cryptology. Volume 3, Issue 1, Pages 121–185, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: 10.1515/gcc.2011.006, May 2011
- Published Online:
In this paper we study the structure of random van Kampen diagrams over finitely presented groups. Such diagrams have many remarkable properties. In particular, we show that a random van Kampen diagram over a given group is hyperbolic, even though the group itself may not be hyperbolic. This allows one to design new fast algorithms for the Word Problem in groups. We introduce and study a new filling function, the depth of van Kampen diagrams, – a crucial algorithmic characteristic of null-homotopic words in the group.
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