  In this note, we explore the notion of hyperbolicity of topologically
finitely generated profinite groups. Some applications to diophantine geometry
are suggested and we try to reformulate certain problems in diophantine
geometry in terms of hyperbolic profinite groups. Then, we introduce many
occasions in which Galois groups are free profinite and try to explore
implications of this condition in the world of diophantine geometry. In
particular, we prove that, Grothendieck's "section conjecture" plus
Shafarevich's "freeness conjecture" imply that hyperbolic curves have
infinitely many solutions over the maximal abelian extension of a global field.
This makes Mordell's conjecture, which was proved by Faltings, more
interesting.
