Brady, N. Ciobanu, Laura Martino, A. Rourke, S. O. 2009 Let <!--l. 11--><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" > <mi >G</mi></math> be a group that acts freely on a <!--l. 11--><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" ><mi >Λ</mi> </math>-tree, where <!--l. 11--><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" ><mi >Λ</mi></math> is an ordered abelian group, and let <!--l. 11--> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" ><mi >x</mi> <mo class="MathClass-punc">,</mo><mi >y</mi><mo class="MathClass-punc">, </mo><mi >z</mi></math> be elements in <!--l. 11--><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" ><mi >G</mi> </math>. We show that if <!--l. 11--><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" ><msup><mrow ><mi >x</mi></mrow><mrow ><mi >p</mi></mrow></msup ><msup><mrow ><mi >y</mi> </mrow><mrow ><mi >q</mi></mrow></msup > <mo class="MathClass-rel">=</mo> <msup><mrow ><mi >z</mi></mrow><mrow ><mi >r</mi></mrow></msup ></math> with integers <!--l. 11--><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" ><mi >p</mi></math>, <!--l. 11--><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" ><mi >q</mi></math>, <!--l. 11--><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" ><mi >r</mi> <mo class="MathClass-rel">≥</mo> <mn>4</mn></math>, then <!--l. 11--> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" ><mi >x</mi> </math>, <!--l. 11--><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" ><mi >y</mi></math> and <!--l. 11--><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" ><mi >z</mi></math> commute. As a result, the one-relator groups with <!--l. 11--><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" ><msup><mrow ><mi >x</mi></mrow><mrow ><mi >p</mi></mrow></msup ><msup><mrow ><mi >y</mi> </mrow><mrow ><mi >q</mi></mrow></msup > <mo class="MathClass-rel">=</mo> <msup><mrow ><mi >z</mi></mrow><mrow ><mi >r</mi></mrow></msup ></math> as relator, are examples of hyperbolic and CAT(<!--l. 11--><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" > <mo class="MathClass-bin">−</mo> <mn>1</mn></math>) groups which do not act freely on any <!--l. 11--><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" ><mi >Λ</mi></math>-tree. application/pdf https://folia.unifr.ch/global/documents/300874 https://folia.unifr.ch/documents/300874/files/ciobani_egf.pdf eng info:eu-repo/semantics/altIdentifier/doi/10.1090/S0002-9947-08-04639-4 info:eu-repo/semantics/openAccess License undefined Transactions of the American Mathematical Society. - 2009, vol. 361, no. 1, p. 223-236 info:eu-repo/classification/udc/51 The equation <span class="cmmi-10">x</span>^{<span class="cmmi-7">p</span>}<span class="cmmi-10">y</span>^{<span class="cmmi-7">q</span>} = <span class="cmmi-10">z</span>^{<span class="cmmi-7">r</span>} and groups that act freely on Λ-trees http://purl.org/coar/resource_type/c_6501