Growth of automorphism groups of relatively free groups (Q2731593)

Let \(\mathbf V\) be a variety of groups such that the relatively free groups of \(\mathbf V\) are residually nilpotent. New definitions of the growth function, the growth and the Gelfand-Kirillov dimension of the automorphism group of a relatively free group \(F_n({\mathbf V})\) in \(\mathbf V\) are introduced. As usual tame automorphism means an automorphism of a relatively free group \(F_n({\mathbf V})\) induced by some automorphism of an absolutely free group \(F_n\). The authors prove that, under some natural restrictions, the growth of the group of tame automorphisms of \(F_n({\mathbf V})\) is equal to the growth of the automorphism group of \(F_n({\mathbf V})\), and coincides with the growth of the Lie algebra over \(\mathbb{Q}\) associated with \(F_n({\mathbf V})\).