Automorphisms in varieties of groups (Q1345916)

If \(N\) is a characteristic subgroup of the group \(G\), then each automorphism of \(G\) induces an automorphism on \(G/N\) and so there is a homomorphism \(\pi:\text{Aut}(G)\to\text{Aut}(G/N)\). Thus if \(V\) is a variety of groups, \(V(F_n)\) the verbal subgroup corresponding to \(V\) and \(F_n(V)\cong F_n/V(F)\) the relatively free group of \(V\) of rank \(n\), then there is the homomorphism \(\pi:\text{Aut}(F_n)\to\text{Aut}(F_n(V))\). The automorphisms of \(F_n(V)\) which belong to the image of \(\pi\) are called tame automorphisms of \(F_n(V)\). The problem, which has drawn recently considerable attention, is for which varieties \(V\) and integers \(n\) the group \(F_n(V)\) has non- tame automorphisms. After giving some examples, the authors restrict themselves to considering only IA-automorphisms, i.e. automorphisms which induce the identity automorphism on the derived factor group \(G/G'\). Their main result says that: (i) If \([x_1,x_2,x_1]\) is a law of the variety \(V\), then every IA-automorphism of \(F_n(V)\) is tame and in fact is induced by an IA-automorphism of \(F_n\). (ii) If \([x_1,x_2,x_1]\) is not a law of \(V\) and \(V\) is a variety which is either locally finite or locally nilpotent, then \(F_n(V)\) has non-tame IA- automorphisms for \(n\geq 2\) and one of them is given explicitly in the latter case.