I asked some time ago:

Let w(x,y) be a word in x and y. Let x and y range among elements of SL_n(K), K a field.

For which w is the map given by

y -> w(x,y)

not surjective for x generic? 

It seems that the map is surjective for most w one can try. At the same time, as I said back when I first asked the question, the map y -> w(x,y) is not surjective for w(x,y) = y x y^{-1}: the image of the map is contained in the conjugacy class of x.

By the same reasoning, the map y -> w(x,y) is not surjective for x generic when w is of the form

w(x,y) =  v(x, u(x,y)^{-1} x u(x,y)),                             (*)

where v and u are some words. The question can then be made precise: are all examples of this form? That is: is it the case that, for all words w(x,y) not of the special form (*), the map y->w(x,y) is surjective for x generic?

I would be extremely interested in the correct answer, even in the case n=3. I strongly suspect that the answer is "yes", at least for n=2.

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[The most obvious approach may be to take derivatives at the origin. However, while this often proves that a map is surjective, it does not prove that a map is not surjective -
and in this problem it leaves too many candidates of possible words w for which the map y->w(x,y) might not be surjective but probably really is.]