# Groups, Geometry, and Dynamics

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**Volume 6, Issue 3, 2012, pp. 409–439**

**DOI: 10.4171/GGD/162**

Published online: 2012-08-16

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

^{[1]}, Shelly Garion

^{[2]}and Fritz Grunewald

^{[3]}(1) Bar-Ilan University, Ramat Gan, Israel

(2) Universität Münster, Germany

(3) Heinrich-Heine-Universität, Düsseldorf, Germany

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 $\mathrm{SL}(2,q)$.

*Keywords: *Engel words, special linear group, arithmetic dynamics, periodic points, finite fields, trace map

Bandman Tatiana, Garion Shelly, Grunewald Fritz: On the surjectivity of Engel words on PSL(2,`q`). *Groups Geom. Dyn.* 6 (2012), 409-439. doi: 10.4171/GGD/162