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Let $F$ be a free group on 2 generators and $G = \operatorname{SL}(d,\mathbb{C})$. A word $w \in F$ induces the word map $\mathrm{ev}_w: G \times G \to G$.

Can we find some (generic) conditions on $A,B \in G$ such that, for all $w \in F$, the differential of $\mathrm{ev}_w$ is surjective at $(A,B)$ ?

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  • $\begingroup$ Are you asking whether it's generic, or whether one can provide explicit generic conditions (while you know it's generic)? $\endgroup$
    – YCor
    Oct 23, 2020 at 11:15
  • $\begingroup$ I'm asking whether it's generic, but an explicit generic condition would be even better. $\endgroup$
    – FMB
    Oct 23, 2020 at 11:21
  • $\begingroup$ So, this is equivalent to asking whether $\mathrm{ev}_w$ is essential for every $w$ (that is, has Zariski-dense image, which by standard algebraic geometry is the same as having image containing a dense Zariski-open subset). $\endgroup$
    – YCor
    Oct 23, 2020 at 13:18
  • $\begingroup$ I believe that a theorem of Borel says that $\text{ev}_w$ is dominant for every non trivial $w$ (see arxiv.org/abs/math/0211302 ). Could you explain how this is equivalent ? $\endgroup$
    – FMB
    Oct 23, 2020 at 15:07
  • $\begingroup$ If $f:X\to Y$ is dominant ($X,Y$ irreducible complex smooth) there's a Zariski-dense open $U$ subset of $X$ on which the differential has maximal rank $r$, and $U\to Y$ is still dominant. At the (metric) neighborhood of any point of $U$ there's a neighborhood whose image is locally (up to local holomorphic change) a copy of $\mathbf{C}^r$ in $\mathbf{C}^{\dim(Y)}$, so if $r<\dim(Y)$ the image is meager, thus can't contain a Zariski-open subset of $Y$. So $r=\dim(Y)$, i.e. the $df(x)$ is surjective for every $x\in U$. $\endgroup$
    – YCor
    Oct 23, 2020 at 15:36

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Since a theorem of Borel (which you quote in the comment) tells you that this regular map is dominant for every $w$, its differential is surjective on a Zariski-dense open subset $U_w$.

Hence there intersection (over all $w\neq 1$) of these subsets $U_w$ is a $\mathrm{G}_\delta$-dense subset of full measure.

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