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My limited knowledge so far is that some groups have been constructed geometrically using the theory of covering spaces, then applying Hilbert irreducibility.

Is there a deeper way in which inverse Galois theory is connected to, for example, Grothendieck's algebraic geometry and the study of etale fundamental groups?

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    $\begingroup$ I think Galois actions on Tate modules and etale cohomology groups have been used to solve new cases of the inverse Galois problem. $\endgroup$
    – Will Sawin
    Commented Feb 16, 2022 at 22:59
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    $\begingroup$ There was an approach by Emmy Noether that depended on rationality (over $\mathbb{Q}$) of some quotients of affine space by finite group actions. However, it was proved by Saltman (and Bogomolov, et al.) that some of these quotients are not rational (in fact, not even stably rational). $\endgroup$
    – Jason Starr
    Commented Feb 16, 2022 at 23:13

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