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The equation $ x^py^q=z^r$ and groups that act freely on $ \Lambda$-trees


Authors: N. Brady, L. Ciobanu, A. Martino and S. O Rourke
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
Published electronically: August 19, 2008
MathSciNet review: 2439405
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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
Email: laura.ciobanu@unifr.ch

A. Martino
Affiliation: Department of Mathematics, Universitat Politècnica de Catalunya, 08860 Castelldefels, Spain
Email: Armando.Martino@upc.edu

S. O Rourke
Affiliation: Department of Mathematics, Cork Institute of Technology, Cork, Ireland
Email: Shane.ORourke@cit.ie

DOI: https://doi.org/10.1090/S0002-9947-08-04639-4
Keywords: Free actions, $\Lambda $-trees, hyperbolic groups, CAT($-1$).
Received by editor(s): December 6, 2006
Published electronically: August 19, 2008
Article copyright: © Copyright 2008 American Mathematical Society

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