I will speak about a geometric method, based on the classical trace map, for investigating word maps on groups PSL(2, q) and SL(2, q). In two different papers (with F. Grunewald, B. Kunyavskii, and Sh. Garion, F. Grunewald, respectively) this approach was applied to the following problems.
1. Description of Engel-like sequences of words in two variables which characterize finite
solvable groups. The original problem was reformulated in the language of verbal dynamical
systems on SL(2). This allowed us to explain the mechanism of the proofs for known
sequences and to obtain a method for producing more sequences of the same nature.
2. Investigation of the surjectivity of the word map defined by the n-th Engel word
[[[X, Y ], Y ], . . . , Y ] on the groups PSL(2, q) and SL(2, q). Proven was that for SL(2, q), this
map is surjective onto the subset SL(2, q) $\setminus$ {−id} $\subset$ SL(2, q) provided that q $\ge q_0(n)$ is
sufficiently large. If $n\le 4$ then the map was proven to be surjective for all PSL(2, q).