Gaps in probabilities of satisfying some commutator-like identities

Abstract

We show that there is a positive constant $\delta <1$ such that the probability of satisfying either the $2$-Engel identity $[X_1,X_2,X_2]=1$ or the metabelian identity $[[X_1,X_2],[X_3,X_4]]=1$ in a finite group is either $1$ or at most $\delta$.

23 pages; referees' comments, new Lemma 4.1.1, correcting and simplifying 4.4.2 and 4.6.2