On finitely generated profinite groups I: strong completeness and   uniform bounds
  We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite groups: let $w$ be a `locally finite' group word and $d\in\mathbb{N}$. Then there exists $f=f(w,d)$ such that in every $d$-generator finite group $G$, every element of the verbal subgroup $w(G)$ is equal to a product of $f$ $w$-values.   An analogous theorem is proved for commutators; this implies that in every finitely generated profinite group, each term of the lower central series is closed.   The proofs rely on some properties of the finite simple groups, to be established in Part II. 
Nikolay Nikolov, Dan Segal