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Let $G = \mathrm{SL}_n$ (say). Then, for $g$ regular semisimple, the conjugacy class $\mathrm{Cl}(g)$ has dimension $= n^2-n$ as a variety, and so $\mathrm{Cl}(g)^{n+1}$ and $G^n$ have the same dimension. Is it possible to build a word map

$$\mathrm{Cl}(g)^{n+1} \to G^n$$ $$(g_1,\dotsc,g_{n+1})\mapsto (w_1(\vec{g}),w_2(\vec{g}),w_3(\vec{g}),\dotsc)$$

that is almost-injective, that is, such that the preimage of a typical point in the image has a bounded number of distinct elements?

It is possible to answer this question (with "yes") using algebraic geometry, if one accepts a pretty terrible dependence of the bound on $n$. I am more interested in what can be obtained by means of character theory, say, with the objective being a good dependence of the bounds (the bound on the preimage, that is, or the constants implicit in the definition of "typical") on $n$.

(Feel free to consider $\mathrm{Cl}(g)^{d (n+1)}$ and $G^{d n}$ instead of $\mathrm{Cl}(g)^{n+1}$ and $G^n$; the important thing is that the dimensions match.)

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  • $\begingroup$ Could you explain (possibly in an answer) the proof using algebraic geometry? $\endgroup$
    – LSpice
    Commented Apr 16 at 1:36
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    $\begingroup$ @LSpice Sure. First, construct a dominant map. (Here $(g_1,g_2,\dotsc,g_{n+1})\mapsto (g_1 g_2^{-1},\dotsc,g_n g_{n+1}^{-1})$ will do nicely.) The preimage of a typical point (a "generic" point in the classical sense) is $0$-dimensional, and so the number of points on it is bounded by its degree as a variety. Bounding that degree by a quantity independent of the ground field (but not independent of $n$) is not particularly hard. $\endgroup$
    – H A Helfgott
    Commented Apr 16 at 5:19
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    $\begingroup$ (That the preimage of a generic point under a dominant map is $0$-dimensional is something that can be found, say, towards the end of Chapter I in Mumford's Red Book; over arbitrary fields, it has to date to Chevalley at the latest.) $\endgroup$
    – H A Helfgott
    Commented Apr 16 at 5:35
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    $\begingroup$ About your definition of "almost injective": a constant map satisfies this (the preimage of a generic point has cardinal 0). You probably don't want this. $\endgroup$
    – YCor
    Commented Apr 16 at 9:04
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    $\begingroup$ So if I understand correctly, eventually you want such a dominant map, with a reasonable bound (w.r. to $n$) on the cardinal of the preimage of a generic point? $\endgroup$
    – YCor
    Commented Apr 16 at 10:52

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