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

Abstract

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 ≥ 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 PSL(2, q) when q ≥ q 0 (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).

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

“…This implies the following results, obtained in [9]. In certain cases, is always surjective on PSL(2 ).…”

“…It was proved in [41] that the word = 2 2 ∈ F 2 , the commutator word = [ ] ∈ F 2 , as well as the words = [ 1 ] ∈ F , -fold commutators in any arrangement of brackets, are almost equidistributed on the family of finite simple non-abelian groups. The case G = SL(2 ) was studied in some more detail in [8,9,14], see Section 6.…”

“…It is proven in [9] that a matrix ∈ SL(2 ) with tr = can be written as = +1 ( ) for some ∈ SL (2 ) In [9] it was shown that the curve C is absolutely irreducible over any finite field F . In addition, its genus satisfies the inequality (C ) ≤ 2 ( − 1) + 1, and it has at most δ = 5 · 2 punctures.…”

“…It was therefore conjectured by Shalev that a similar result also holds for iterated commutator (so-called Engel words). The study of this case started in [9] (for G = SL(2 ), see subsection 6.1). The words of the form = ∈ F 2 have also attracted special interest.…”

“…In [9] the particular case of Engel words in the groups PSL(2 ) and SL(2 ) was analyzed using the trace map method, in an attempt to prove Conjecture 5.18 for PSL (2 ). The main idea was to check the surjectivity of the trace map on A 3 (F ) instead of the surjectivity of the word map on SL(2 ).…”