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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 -> 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 -> 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->y x^n y^{-1}), and, as we just said, y-> 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^n x^b (v(x,y))^{-1}, v(x,y))^{-1} x^c, where v is some word and n is a,b,c are some integerintegers. (This seems to me a sensible guess, though I would actually be quite glad if it weren't true.)
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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 -> 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}. 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) = v(x,y) x x^n (v(x,y))^{-1}, where v is some word and n is some integer. (This seems to me a sensible guess, though I would actually be quite glad if it weren't true.)
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When is a map given by a word surjective?

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 -> 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}. 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) = v(x,y) x (v(x,y))^{-1}, where v is some word. (This seems to me a sensible guess, though I would actually be quite glad if it weren't true.)