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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On torsion-free groups with finite regular file bases
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by Alexey Muranov PDF
Trans. Amer. Math. Soc. 359 (2007), 3609-3645 Request permission

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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Additional Information
  • 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
  • Email: muranov@math.univ-lyon1.fr
  • 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.
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 3609-3645
  • MSC (2000): Primary 20F05; Secondary 20F06
  • DOI: https://doi.org/10.1090/S0002-9947-07-04256-0
  • MathSciNet review: 2302509