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 PSL(2,q) and SL(2,q). For SL(2,q) we show that this map is surjective onto the subset SL(2,q)\{-id} ⊂ SL(2,q) provided that q > q0(n) is sufficiently large. Moreover, we give an estimate for q0(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 PSL(2,q) when q > q0(n). By using a computer, we sharpen this result and show that for any n < 4 the corresponding map is surjective for all the groups 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 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 SL(2,q).