We investigate the surjectivity of the word map defined by the -th Engel word on the groups and . For we show that this map is surjective onto the subset provided that is sufficiently large. Moreover, we give an estimate for . We also present examples demonstrating that this does not hold for all . We conclude that the -th Engel word map is surjective for the groups when . By using a computer, we sharpen this result and show that for any the corresponding map is surjective for all the groups . This provides evidence for a conjecture of Shalev regarding Engel words in finite simple groups. In addition, we show that the -th Engel word map is almost measure-preserving for the family of groups , with 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 .
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. 409–439DOI 10.4171/GGD/162