On the surjectivity of Engel words on PSL(2,$q$)

Abstract

We investigate the surjectivity of the word map defined by the $n$-th Engel word on the groups $\mathrm{PSL}(2,q)$ and $\mathrm{SL}(2,q)$. For $\mathrm{SL}(2,q)$ we show that this map is surjective onto the subset $\mathrm{SL}(2,q)\setminus \{-\mathrm{id}\}\subset \mathrm{SL}(2,q)$ provided that $q\ge q_0(n)$ is sufficiently large. Moreover, we give an estimate for $q_0(n)$. We also present examples demonstrating that this does not hold for all $q$. We conclude that the $n$-th Engel word map is surjective for the groups $\mathrm{PSL}(2,q)$ when $q\ge q_0(n)$. By using a computer, we sharpen this result and show that for any $n\le 4$ the corresponding map is surjective for all the groups $\mathrm{PSL}(2,q)$. This provides evidence for a conjecture of Shalev regarding Engel words in finite simple groups. In addition, we show that the $n$-th Engel word map is almost measure-preserving for the family of groups $\mathrm{PSL}(2,q)$, with $q$ odd, answering another question of Shalev.

Our techniques are based on the method developed by Bandman, Grunewald and Kunyavskii for verbal dynamical systems in the group $\mathrm{SL}(2,q)$.