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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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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

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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.

Keywords.: Word problem; finitely presented groups; search algorithms; van Kampen diagram; hyperbolic diagrams; Todd–Coxeter algorithm

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