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)

Tatiana Bandman[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