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).