The equation 𝑥^{𝑝}𝑦^{𝑞}=𝑧^{𝑟} and groups that act freely on Λ-trees

Brady, N.; Ciobanu, L.; Martino, A.; O Rourke, S.

doi:10.1090/S0002-9947-08-04639-4

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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