# On the surjectivity of Engel words on PSL(2,<var>q</var>)

### Tatiana Bandman

Bar-Ilan University, Ramat Gan, Israel### Shelly Garion

Universität Münster, Germany### Fritz Grunewald

Heinrich-Heine-Universität, Düsseldorf, Germany

## 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 $q \geq q_0(n)$ 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 $q \geq q_0(n)$. By using a computer, we sharpen this result and show that for any $n \leq 4$ 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 $\mathrm{SL}(2,q)$.

## Cite this article

Tatiana Bandman, Shelly Garion, Fritz Grunewald, On the surjectivity of Engel words on PSL(2,<var>q</var>). Groups Geom. Dyn. 6 (2012), no. 3, pp. 409–439

DOI 10.4171/GGD/162