  Fontaine-Mazur Conjecture is one of the core statements in modern arithmetic
geometry. Several formulations were given since its original statement in 1993,
and various angles have been adopted by numerous authors to try to tackle it.
Boston's seminal paper in 1992 gave a range of purely group-theoretic methods
rather than representation-theoretic ones to prove some special cases of this
conjecture. Such methods have been later successfully carried on by Maire and
his co-authors, and brings different informations on the objects involved in
the conjecture. This survey article aims to review what is known in this
direction and to present some interesting related questions the authors work
on.
