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

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) .