A note on Lyndon properties in one-relator groups.
The authors call the property in a group \(G\) that \(x^2y^2z^2=1\) for \(x,y,z\in G\) implies that \(\langle x,y,z\rangle\) is cyclic `Lyndon's property'. Lyndon showed this property for free groups. Let \(F_3\) be a free group of rank 3 with basis \(\{x_1,x_2,x_3\}\). A word \(w=w(x_1,x_2,x_3)\) is called `regular' if there is no Nielsen transformation from \(\{x_1,x_2,x_3\}\) to a system \(\{y_1,y_2,y_3\}\) with \(w\in\langle y_1,y_2\rangle\). The word \(w\) is called `quadratic' if each \(x_i\) in \(w\) occurs exactly twice, each time as \(x_i\) or \(x_i^{-1}\). In the present paper it is shown: Theorem 1. Let \(G\) be the one-relator group \(G=\langle a,b,c,\dots;\;R^m\rangle\) with \(m\geq 3\) odd and \(R\) a cyclically reduced word, not a proper power in the free group on \(a,b,c,\dots\). Let \(w(x_1,x_2,x_3)\) be a regular quadratic word in the free group \(F\) on \(x_1,x_2,x_3\) and let \(\varphi\colon F\to G\) be a homomorphism from \(F\) into \(G\) with \(\varphi (x_i)=u_i\) for \(i=1,2,3\). If \(w(u_1,u_2,u_3)=1\) in \(G\) than the subgroup \(\langle u_1,u_2,u_3\rangle\) is cyclic. In particular, \(G\) satisfies the Lyndon property.
scientific article; zbMATH DE number 5911446
scientific article