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} subset of SL(2, q) provided that q >= qo(n) is sufficiently large. Moreover, we give an estimate for qo (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 >= qo(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.