Recognizing automorphisms of the free groups
If \(F\) is a free group generated freely by the system \(\{x_ i\}_{i \in I}\), where \(I\) is finite or countable infinite, then each mapping \(\varphi\) taking \(x_ i\) to \(y_ i\) of \(F\), for each \(i \in I\), gives an endomorphism of \(F\). This paper is concerned with the problem of deciding whether or not this endomorphism \(\varphi\) is actually an automorphism. There are the classical algorithms by Nielsen and Whitehead and also the more recent ``inverse function theorem'' by \textit{J. S. Birman} [Proc. Am. Math. Soc. 41, 634-638 (1973; Zbl 0274.20032)], which are applied in the case where the \(y_ i\) are given explicitly. In the case of two generators there is Nielsen's ``commutator test'' which says that \(\varphi\) is an automorphism if \([y_ 1,y_ 2]\) is conjugate to \([x_ 1,x_ 2]\) or its inverse. The main result of the present paper gives a generalization of Nielsen's test. More precisely it is proved: Let \(R\) be an arbitrary subgroup of a free group \(F\). Then: (a) generators \(x_ 1,\dots,x_ n\) of \(F\) belong to \(R\) if and only if the element \(u_ 1(n)=[x_ 1,\dots,x_ n]\) belongs to \(\gamma_ n(R)\); (b) for \(n=2m\), generators \(x_ 1,\dots,x_ n\) of \(F\) belong to \(R\) if and only if the element \(u_ 2(2m)=[x_ 1,x_ 2]\dots [x_{2m- 1},x_{2m}]\) belongs to \(R^ 2\); (c) for an arbitrary \(k \geq 2\), generators \(x_ 1,\dots,x_ n\) of \(F\) belong to \(R\) if and only if the element \(u_ 3(n,k)=x^ k_ 1 \dots x^ k_ x\) belongs to \(R' R^ k\). One of the corollaries states that if \(\varphi\) is an endomorphism of \(F=F_ n\), \(n \geq 2\) and \(u\) is one of the words \(u_ 1(n)\), \(u_ 2(2m)\), \(u_ 3(n,k)\) then \(\varphi\) is an automorphism if and only if \(\varphi(u)=\alpha(u)\) for some \(\alpha \in \text{Aut }F_ n\). It is also proved that if \(N\) is a fully invariant subgroup of the free group \(F=F_ n\), \(n=2m\), \(G=F/[N',F]\) and \(\varphi\) an endomorphism of \(G\) such that \(\varphi(u_ 2(2m))=\alpha(u_ 2(2m))\) for some \(\alpha \in \text{Aut }G\), then \(\varphi\) is onto. For the proofs the author is working in the group ring \(\mathbb{Z} F\) utilizing Fox's free derivatives.
scientific article; zbMATH DE number 590065