Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

Abstract

A right Engel sink of an element $g$ of a group $G$ is a set $\mathcal{R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to $\mathcal{R}(g)$. (Thus, $g$ is a right Engel element precisely when we can choose $\mathcal{R}(g)=\{1\}$.) We prove that if a profinite group $G$ admits a coprime automorphism $\phi$ of prime order such that every fixed point of $\phi$ has a finite right Engel sink, then $G$ has an open locally nilpotent subgroup. A left Engel sink of an element $g$ of a group $G$ is a set $\mathcal{E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to $\mathcal{E}(g)$. (Thus, $g$ is a left Engel element precisely when we can choose $\mathcal{E}(g)=\{1\}$.) We prove that if a profinite group $G$ admits a coprime automorphism $\phi$ of prime order such that every fixed point of $\phi$ has a finite left Engel sink, then $G$ has an open pronilpotent-by-nilpotent subgroup.