On the surjectivity of Engel words on \(PSL(2,q)\)
The paper presents several results regarding the \(n\)-th Engel word map on \(\text{PSL}(2,q)\) and \(\text{SL}(2,q)\), where \(n\) and \(q\) range through the natural numbers. The Engel word maps are recursively defined generalizations of the commutator word map, and the results in the paper are extensions of previously know results about the commutator word map. The authors note that a key difficulty in moving beyond the commutator word map is that there is no (known) quantity for Engel words that is analogous to the character sum for the commutator word. The main result is that, like the commutator word map, the \(n\)-th Engel word map on \(\text{PSL}(2,q)\) is surjective in many cases and should be expected to always be surjective. It is also shown, however, that the \(n\)-th Engel word map on \(\text{SL}(2,q)\) is not even almost surjective in infinitely many cases (in contrast to the situation for the commutator map). The paper also includes results about almost equidistribution and almost measure-preservation for the Engel word maps on \(\text{PSL}(2,q)\).NEWLINENEWLINEThe main tool used in the paper is a so-called trace map on an affine space whose definition is such that surjectivity of the \(n\)-th Engel word map on \(\text{PSL}(2,q)\) corresponds to surjectivity of the \(n\)-th iterate of the trace map. Thus the results are achieved via the dynamics of the trace map. The proof of the main result relies on a nice accounting of points on certain affine curves defined over finite fields.
scientific article; zbMATH DE number 6088867