Non-singular word maps for linear groups

We study the word image of words with constants in $\mathrm{G}\mathrm{L}(V)$ and show that it is large provided the word satisfies some natural conditions on its length and its critical constants. There are various consequences: We prove that for every $l\ge 1$, there are only finitely many pairs $(n,q)$ such that the length of the shortest non-singular mixed identity ${\mathrm{P}\mathrm{S}\mathrm{L}}_n(q)$ is bounded by $l$. We generalize the Hull–Osin dichotomy for highly transitive permutation groups to linear groups over finite fields. Finally, we show that the rank limit of ${\mathrm{G}\mathrm{L}}_n(q)$ for $q$ fixed and $n\to \mathrm{\infty }$ is mixed identity free.