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


We investigate the surjectivity of the word map defined by the nn-th Engel word on the groups PSL(2,q)\mathrm{PSL}(2,q) and SL(2,q)\mathrm{SL}(2,q). For SL(2,q)\mathrm{SL}(2,q) we show that this map is surjective onto the subset SL(2,q){id}SL(2,q)\mathrm{SL}(2,q)\setminus\{-\mathrm{id}\}\subset \mathrm{SL}(2,q) provided that qq0(n)q \geq q_0(n) is sufficiently large. Moreover, we give an estimate for q0(n)q_0(n). We also present examples demonstrating that this does not hold for all qq. We conclude that the nn-th Engel word map is surjective for the groups PSL(2,q)\mathrm{PSL}(2,q) when qq0(n)q \geq q_0(n). By using a computer, we sharpen this result and show that for any n4n \leq 4 the corresponding map is surjective for all the groups PSL(2,q)\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 nn-th Engel word map is almost measure-preserving for the family of groups PSL(2,q)\mathrm{PSL}(2,q), with qq 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 SL(2,q)\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