Commutators, centralizers, and strong conciseness in profinite groups

A group G is said to have restricted centralizers if for each (Formula presented.) the centralizer (Formula presented.) either is finite or has finite index in G. 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 (Formula presented.), where (Formula presented.). Here, (Formula presented.) denotes the set of prime divisors of the order of (Formula presented.). It is shown that such a group necessarily has an open nilpotent subgroup. We use this result to deduce that (Formula presented.) is finite if and only if the cardinality of the set of uniform k-step commutators in G is less than (Formula presented.).