Bounded Engel elements in residually finite groups

Abstract

Let q be a prime. Let G be a residually finite group satisfying an identity. Suppose that for every x ∈ G there exists a q-power m = m(x) such that the element x m is a bounded Engel element. We prove that G is locally virtually nilpotent. Further, let d, n be positive integers and w a non-commutator word. Assume that G is a d-generator residually finite group in which all w-values are n-Engel. We show that the verbal subgroup w(G) has {d, n, w}bounded nilpotency class.2010 Mathematics Subject Classification. 20F45, 20E26.