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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On torsion-free groups with finite regular file bases

Author: Alexey Muranov
Journal: Trans. Amer. Math. Soc. 359 (2007), 3609-3645
MSC (2000): Primary 20F05; Secondary 20F06
Published electronically: March 7, 2007
MathSciNet review: 2302509
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Abstract: The following question was asked by V. V. Bludov in The Kourovka Notebook in 1995: If a torsion-free group $ G$ has a finite system of generators $ a_{1}$, ..., $ a_{n}$ such that every element of $ G$ has a unique presentation in the form $ a_{1}^{k_{1}}\cdots a_{n}^{k_{n}}$ where $ k_{i}\in \mathbb{Z}$, is it true that $ G$ is virtually polycyclic? The answer is ``not always.'' A counterexample is constructed in this paper as a group presented by generators and defining relations.

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Alexey Muranov
Affiliation: Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, Tennessee 37240–0001
Address at time of publication: Institut Camille Jordan, Université Lyon 1, 43 blvd du 11 novembre 1918, 69622 Villeurbanne cedex, France

Keywords: Group presentation, van Kampen's lemma, diagram with selection, S-diagram, virtually polycyclic group, boundedly generated group, file basis
Received by editor(s): March 4, 2005
Published electronically: March 7, 2007
Additional Notes: This work was supported in part by the NSF grant DMS 0245600 of Alexander Ol’shanskiy and Mark Sapir.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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