9 - Braiding Action and Weak Rigidity

Summary

Weakly rigid (Galois) extensions of ℂ(x) are those that are uniquely determined by their ramification type T. If such an extension L/ℂ(x) is even rigid then its minimal field of definition equals a certain field κ_T that is quite explicitly given in terms of the ramification type T. Thus G = G(L/ℂ(x)) occurs as a Galois group over κ_T(x), and this yields the rigidity criteria from Chapter 3.

In the weakly rigid case, we still get a Galois realization over the field κ_T(x); not for G, however, but for a certain group G_T of automorphisms of G (that contains the inner automorphisms); see Proposition 9.2. This raises the question of computing G_T, or equivalently, the image of G_T in Out(G). In general, this is a very difficult question, essentially equivalent to the regular version of the Inverse Galois Problem (since every finite extension of ℂ(x) embeds into a weakly rigid one). A partial answer can be given in the “generic” case that the branch points of L/ℂ(x) are algebraically independent. Then the image of G_T in Out(G) has a normal subgroup Δ that can be described purely combinatorially in terms of the ramification type T. This description involves the braiding action on generating systems of G (Theorem 9.5).

The proof of this theorem is given in Chapter 10. It requires topological and analytic methods, as for Riemann's existence theorem.