Gelfand?Kirillov Dimension for Automorphism Groups of Relatively Free Algebras

Abstract

Using ideas of our recent work on automorphisms of residually nilpotent relatively free groups, we introduce a new growth function for subgroups of the automorphism groups of relatively free algebras Fn(V) over a field of characteristic zero and the related notion of Gelfand-Kirillov dimension, and study their behavior. We prove that, under some natural restrictions, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fn(V) is equal to the Gelfand-Kirillov dimension of the algebra Fn(V). We show that, in some cases, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fn(V) is smaller than the Gelfand-Kirillov dimension of the whole automorphism group, and calculate the Gelfand-Kirillov dimension of the automorphism group of Fn(V) for some important varieties V.