Test elements for endomorphisms of free groups and algebras
This paper is about ``test elements'' for endomorphisms of free groups and algebras. An element \(u\), of the free group \(F_n\) of rank \(n\), is a test element if whenever \(\varphi(u)= u\) for an endomorphism \(\varphi\), then \(\varphi\) is actually an automorphism of \(F_n\). The author discusses what is known about these elements. But the main object of the paper is another approach to the problem. If \(u\in F_n\) then he considers ``the double Jacobian matrix'' \(D_u= |d_i' (d_j (u)) |_{1\leq i,j\leq n}\), where \(d_i\) is the usual left Fox derivation and \(d_i'\) is the right Fox derivation, defined in an analogous manner. Of course, the derivations have values in the free group ring \(\mathbb{Z} F\). In terms of these matrices he proves the following theorem: Let \(\varphi\) be an endomorphism of the group \(F= F_n\). Then \(\varphi\) is an automorphism: (i) if and only if the matrix \(D_{\varphi(u)}\) is invertible over \(\mathbb{Z} F\) with \(u=[x_1, x_2]\dots[x_{n-1}, x_n]\), \(n\) even; (ii) if and only if the natural image over \(\mathbb{Z}_2 F\) of the matrix \(D_{\varphi (u)}\) is invertible over \(\mathbb{Z}_2 F\) with \(u= x^2_1 x^2_2 \dots x^2_n\). In another direction he considers elements of \(\mathbb{Z} F\) which can be used to distinguish any two different endomorphisms of \(F\) by means of their value on these elements. For example he proves: Let \(u= \sum_{1\leq i\leq n} p_i (x_i- 1)^2\) with different odd integers \(p_i\) and let \(\varphi\) and \(\psi\) be two endomorphisms of \(F\) such that \(\varphi(u)= \psi(u)\). Then \(\varphi= \psi\). He also proves similar results in the case of associative or Lie algebras.
scientific article; zbMATH DE number 836170