Commutators, centralizers, and strong conciseness in profinite groups

A group G is said to have restricted centralizers if for each g∈G$g \in G$ the centralizer CG(g)$C_G(g)$ 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 [x1,⋯,xk]$[x_1,\dots ,x_k]$, where π(x1)=π(x2)=⋯=π(xk)$\pi (x_1)=\pi (x_2)=\dots =\pi (x_k)$. Here, π(x)$\pi (x)$ denotes the set of prime divisors of the order of x∈G$x\in G$. It is shown that such a group necessarily has an open nilpotent subgroup. We use this result to deduce that γk(G)$\gamma _k(G)$ is finite if and only if the cardinality of the set of uniform k‐step commutators in G is less than 2ℵ0$2^{\aleph _0}$.