On an article by W. Magnus on the Fricke characters of free groups (Q1570371)

The paper is inspired by the Fricke character of the free group \(\Gamma_{\langle a_1,\dots,a_n\rangle}\) with \(n\) generators \(a_1,\dots,a_n\) which associates to every element \(w\in\Gamma_{\langle a_1,\dots,a_n\rangle}\) a polynomial \(P_w(x_1,\dots,x_n)\) in the traces of the \(2^n-1\) products \(x_{j_1}\cdots x_{j_k}\), \(1\leq j_1<\cdots<j_k\leq n\), with the following property. For any homomorphism \(\phi\colon\Gamma_{\langle a_1,\dots,a_n\rangle}\to\text{SL}_2(\mathbb{C})\) the trace of \(\phi(w)\) is equal to \(P_w(\text{tr}(\phi(a_1)),\dots,\text{tr}(\phi(a_n)))\). \textit{W. Magnus} [Math. Z. 170, 91-103 (1980; Zbl 0433.20033)] elucidated the structure of the ideal of polynomials in \(2^n-1\) variables which vanish on any system of traces of \(\phi(a_{j_1}\cdots a_{j_n})\). Since the Fricke character is the evaluation of the trace of products of generic \(2\times 2\) matrices under the additional assumption that the determinants are equal to 1, the idea of the author of the present paper is to involve methods typical for algebras with polynomial identities. He considers the more general situation of algebras which are algebraic of degree 2 over a field and shows that these algebras are either \(2\times 2\) matrix algebras or algebras with one- or two-dimensional factor algebra modulo the nilpotent of class 3 Jacobson radical. The author makes applications to Fricke characters. He also establishes some relations between generic \(2\times 2\) matrices. Some of them are already known (maybe in another form) in the theory of PI-algebras.