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 the 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).