Commutators, centralizers, and strong conciseness in profinite groups

Abstract

A group 𝐺 is said to have restricted centralizers if for each 𝑔 ∈ 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 , … , 𝑥 𝑘 ], where 𝜋(𝑥 1 ) = 𝜋(𝑥 2 ) = ⋯ = 𝜋(𝑥 𝑘 ). Here, 𝜋(𝑥) denotes the set of prime divisors of the order of 𝑥 ∈ 𝐺. It is shown that such a group necessarily has an open nilpotent subgroup. We use this result to deduce that 𝛾 𝑘 (𝐺) is finite if and only if the cardinality of the set of uniform 𝑘-step commutators in 𝐺 is less than 2 ℵ 0 .