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 $\pi (x_1)=\pi (x_2)=\cdots =\pi (x_k)$. 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 $2^{{\aleph }_0}$