Commutators, centralizers, and strong conciseness in profinite groups

Abstract: A group $G$ is said to have restricted centralizers if for each $g \in G$ the centralizer $C_G(g)$ 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 $[x_1,\dots,x_k]$, where $\pi(x_1)=\pi(x_2)=\dots=\pi(x_k)$. Here $\pi(x)$ denotes the set of prime divisors of the order of $x\in G$. 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 $2^{\aleph_0}$