This review Appears in Recent Developments in Inverse Galois Problem conference, p. 15--32.

Serre didn't use the braid monodromy (rigidity) method, neccessary to achieve anything significant in regular realizations except for the few lucky groups for which three conjugacy classes happen to satisfy the rigidity criterion. Yet, modular curves appear everywhere in his book; this paper makes the connection to braid ridigity through Serre's own exercises. The difference shows almost immediately in considering the realizations of Chevalley groups of rank exceeding one.

Serre's book records just three examples of Chevalley groups of ranks exceeding one having known regular realizations at the time of his book. Two papers of Fried-Voelklein (Arithmetic of Covers section) used the braid monodromy method to turn the R(egular) I(nverse) G(alois) P(roblem) into a purely diophantine matter of rational points on Hurwitz spaces. To make this work for a given group G requires identifying collections of absolutely irreducible Hurwitz space components HG whose rational points would correspond to RIGP solutions for G.

The Conway-Fried-Parker-Voelklein appendix of the Mathematische Annalen paper was a non-explicit method for doing that. The "Alternating Groups and Moduli Space Lifting Invariants" paper shows what it can mean to be very explicit about this. A presentation of the absolute Galois group of Q by known groups (a profree group on one hand, and a product of symmetric groups on the other) was an example of the power of the method (in the 1992 Annals paper). This made obvious the right generalization of Shafarevic's Cyclotomic field conjecture:

(*) An Hilbertian subfield of the algebraic numbers should have profree absolute Galois group if and only its Galois group is projective.

Soon after this Voelklein and Thompson – albeit powerful group theorists – produced high rank Chevalley groups in abundance based on the same method. Locating specific high-dimensional uni-rational Hurwitz spaces was the key here. Examples, and the elementary uses of Riemann's Existence Theorem, abound in H. Voelkein, Groups as Galois Groups, Cambridge Studies in Advanced Mathematics 53 (1996).