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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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The equation $x^py^q=z^r$ and groups that act freely on $\Lambda$-trees
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by N. Brady, L. Ciobanu, A. Martino and S. O Rourke PDF
Trans. Amer. Math. Soc. 361 (2009), 223-236 Request permission

Abstract:

Let $G$ be a group that acts freely on a $\Lambda$-tree, where $\Lambda$ is an ordered abelian group, and let $x, y, z$ be elements in $G$. We show that if $x^p y^q = z^r$ with integers $p$, $q$, $r \geq 4$, then $x$, $y$ and $z$ commute. As a result, the one-relator groups with $x^p y^q = z^r$ as relator, are examples of hyperbolic and CAT($-1$) groups which do not act freely on any $\Lambda$-tree.
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Additional Information
  • N. Brady
  • Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
  • Email: nbrady@math.ou.edu
  • L. Ciobanu
  • Affiliation: Department of Mathematics, University of Fribourg, CH-1700 Fribourg, Switzerland
  • MR Author ID: 797163
  • Email: laura.ciobanu@unifr.ch
  • A. Martino
  • Affiliation: Department of Mathematics, Universitat Politècnica de Catalunya, 08860 Castelldefels, Spain
  • MR Author ID: 646503
  • Email: Armando.Martino@upc.edu
  • S. O Rourke
  • Affiliation: Department of Mathematics, Cork Institute of Technology, Cork, Ireland
  • Email: Shane.ORourke@cit.ie
  • Received by editor(s): December 6, 2006
  • Published electronically: August 19, 2008
  • © Copyright 2008 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 223-236
  • MSC (2000): Primary 20E08, 20F65
  • DOI: https://doi.org/10.1090/S0002-9947-08-04639-4
  • MathSciNet review: 2439405