Demushkin groups of infinite rank in Galois theory

Demushkin groups play an important role in number theory, being the maximal pro-$p$ Galois groups of local fields containing a primitive root of unity of order $p$. In 1996 Labute presented a generalization of the theory for countably infinite rank pro-$p$ groups, and proved that the $p$-Sylow subgroups of the absolute Galois groups of local fields are Demushkin groups of infinite countable rank. These results were extended by Minac & Ware, who gave necessary and sufficient conditions for Demushkin groups of infinite countable rank to occur as absolute Galois groups.

In a joint work with Prof. Nikolay Nikolov, we extended this theory further to Demushkin groups of uncountable rank. Since for uncountable cardinals, there exists the maximal possible number of nondegenerate bilinear forms, the class of Demushkin groups of uncountable rank is much richer, and in particular, the groups are not determined completely by the same invariants as in the countable case.  

Additionally, inspired by the Elementary Type Conjecture by Ido Efrat and the affirmative solution to Jarden's Question, we discuss the possibility of a free product over an infinite sheaf of Demushkin groups of infinite countable rank to be realizable as an absolute Galois group, and give a necessary and sufficient condition when the free product is taken over a set converging to 1.