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 $(A,B)$ ?



Local surjectivity of word maps
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.


