Commutators, centralizers, and strong conciseness in profinite groups

A group.. is said to have restricted centralizers if for each g epsilon G the centralizer C-G(g) either is finite or has finite index in... Shalev showed that a profinite group with restricted centralizers is virtually abelian. We take interest in profinite groups with restricted centralizers of uniform commutators, that is, elements of the form [x1,.,...xk.], where pi (x1) = pi(x2) = ... =pi (xk). Here, pi(x) denotes the set of prime divisors of the order of...... It is shown that such a group necessarily has an open nilpotent subgroup. We use this result to deduce that gamma k(G) is finite if and only if the cardinality of the set of uniform k-step commutators in G is less than 2(N0).