On triangular representations of \(\Aut(F_r)\) (Q1969478)

W. Burnside has proved that finitely generated torsion linear groups over a field of characteristic 0 are finite. If only the generators are of finite order then this result is no longer true, as one can deduct from the group \(\text{SL}(2,\mathbb{Z})\) which is generated by torsion elements. In this paper the author examines the case where some of the elements are of finite order. More precisely the main result of the paper is the following theorem: Let \(\rho\colon F_r\to\text{GL}_n(K)\) be a representation of the finitely generated free group \(F_r\), of rank \(r\), over an arbitrary field \(K\). Assume that for some integer \(m\) which is not a multiple of the characteristic of \(K\) and that for each primitive element \(\alpha\) of \(F_r\), \(\rho(a)\) has order \(m\). Then \(\rho(F_r)\) is triangularisable if, and only if, it is Abelian and of finite order dividing \(m^r\). (An element is primitive if it belongs to a basis, i.e. a free generating set of \(F_r\)). For the proof the authors also employs a result of \textit{R. P. Osborne} and \textit{H. Zieschang} [Invent. Math. 63, 17-24 (1981; Zbl 0451.20023)].