A characterization of invertible trace maps associated with a substitution (Q1304929)

This paper gives a characterization of the invertible endomorphisms (and of the invertible trace maps) of a rank 2 free group. If \(\sigma\) denotes an endomorphism of the free group \(F= \langle a,b\rangle\) generated by \(a,b\) and if \(\Phi\in \Hom(F, \text{SL}(2,\mathbb{C}))\), then there exists a unique polynomial map \(\Phi_\sigma\in \mathbb{Z}^3[x,y,z]\) such that \[ (\operatorname {tr} \Phi(\sigma(a)), \operatorname {tr} \Phi(\sigma(b)), \operatorname {tr} \Phi(\sigma(ab)))= \Phi_\sigma (\operatorname {tr} \Phi(a),\operatorname {tr} \Phi(b), \operatorname {tr} \Phi(ab)), \] where \(\operatorname {tr}A\) stands for the trace of the matrix \(A\), and there exists a unique polynomial \(\varphi_\sigma\) such that \(\mathbb{Z}_0 \Phi_\sigma= \mathbb{Z}\varphi_\sigma\), where \(\mathbb{Z}(x,y,z)= x^2+ y^2+ z^2- xyz-4\). It is known that \(\sigma\) is an automorphism if and only if \(\varphi_\sigma\) reduces to the constant polynomial 1. In the paper under review, it is proved that \(\sigma\) is an automorphism if and only if \(\varphi_\sigma(2,2,z)\) reduces to the constant polynomial 1. Let us note that invertible nonnegative endomorphisms (that is, invertible substitutions) of \(F\) are exactly the Sturmian substitutions [\textit{F. Mignosi} and \textit{P. Séébold}, J. Théor. Nombres Bordx. 5, 221-233 (1993; Zbl 0797.11029)].