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


Let $x$ and $y$ now vary in $\operatorname{SL}\_n(K)$, 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


$$y \mapsto w(x,y)$$


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 $y \mapsto y x^n y^{-1}$ 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 $x\* \mathrm{im}(y\mapsto y x^n y^{-1})$,
and, as we just said, $y\mapsto y x^n y^{-1}$ 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.)




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\*Edit:\* The discussion has now moved to this subsequent question:
[Surjective maps given by words, redux](https://mathoverflow.net/questions/2082/surjective-maps-given-by-words-redux)



When is a map given by a word surjective?
The discussion has now moved to


[Surjective maps given by words, redux](https://mathoverflow.net/questions/2082/surjective-maps-given-by-words-redux)


