TITLE: Infinite simple Galois groups
QUESTION [7 upvotes]: Conjecturally, every finite group is the Galois group of some extension of the rationals. 
This question made me wonder what is known about infinite
simple groups occurring as Galois groups. 
What are the infinite simple groups that are expected to be Galois groups, i.e., profinite? Are they classified? Are there any examples of such extensions?

REPLY [13 votes]: Th notion of simple is not very interesting for profinite groups. The "right" concept is just infinite. We say that a profinite group is just infinite if all its non-trivial normal closed subgroups are of finite index. $SL_2(\mathbb{Z}_p)$ is an example of a just infinite profinite group. Another example coming from Galois theory is the Nottingham group which is an open subgroup of index p-1 in the automorphism group of the field $\mathbb{F}_p((t))$.
There are many examples of just infinite profinite groups. There isn't much general theory except Wilson's dichotomy that they are either Branch Groups (e.g. Grigorchuk group or the Gupta-Sidki group) or they contain an open subgroup which is direct sum of hereditarily just infinte profinite groups (i.e. every open subgroup is also just infinite) and some results of Colin Reid.