NIL-AUTOMORPHISMS OF GROUPS WITH RESIDUAL PROPERTIES

Let G be any group and x an automorphism of G. The automorphism x is said to be nil if, for every g is an element of G, there exists n = n(g) such that [g, (n) x] = 1. If n can be chosen independently of g, we say that x is n-unipotent. A nil (resp. unipotent) automorphism x could also be seen as a left Engel element (resp. left n-Engel element) in the group G < x >. When G is a finite dimensional vector space, groups of unipotent linear automorphisms turn out to be nilpotent, so that one might ask to what extent this result can be extended to a more general setting. In this paper we study finitely generated groups of nil or unipotent automorphisms of groups with residual properties (e. g. locally graded groups, residually finite groups, profinite groups), proving that such groups are nilpotent.