Let $w(x,y)$ be a word in $x$ and $y$.

Let $x$ and $y$ now vary in $SL_n(K)$, where $K$ is a field. (Assume, if you wish, that $K$ is an algebraically complete field of characteristic bigger than a constant.)

I would like to know for which words $w$ the map

$y \rightarrow w(x,y)$

isn't surjective (or even dominant - that is, "almost surjective") for $x$ generic.

It is clear, for example, that the map is surjective for $w(x,y)=xy$, and that it isn't surjective for $w(x,y)= y x y^{-1}$, or for $w(x,y) = y x^n y^{-1}$, $n$ an integer: all elements of the image of $y \rightarrow y x^n y^{-1}$ lie in the same conjugacy class. A moment's thought (thanks, Philipp!) shows that $w(x,y) = x y x^n y^{-1}$ isn't surjective either: its image is just $x* im(y\rightarrow y x^n y^{-1})$, and, as we just said, $y\rightarrow y x^n y^{-1}$ isn't surjective.

I would like to know if the only words $w$ for which the map isn't surjective for $x$ generic are the $w$'s of the form $w(x,y) = x^a v(x,y) x^b (v(x,y))^{-1} x^c$, where $v$ is some word and $a,b,c$ are some integers. (This seems to me a sensible guess, though I would actually be quite glad if it weren't true.)