# Profinite groups with restricted centralizers of commutators

## TL;DR

This paper studies profinite groups where the centralizers of certain word-values are restricted, showing that under these conditions, the associated verbal subgroup is close to abelian, extending previous results on centralizers.

## Contribution

It proves that in profinite groups with restricted centralizers of multilinear commutator word-values, the verbal subgroup is abelian-by-finite, generalizing earlier theorems.

## Key findings

- Verbal subgroup w(G) is abelian-by-finite under restricted centralizers.
- Extends Shalev's theorem to word-values in profinite groups.
- Provides structural insights into profinite groups with restricted centralizer conditions.

## Abstract

A group G has restricted centralizers if for each g in G the centralizer C_G(g) either is finite or has finite index in G. A theorem of Shalev states that a profinite group with restricted centralizers is abelian-by-finite. In the present article we handle profinite groups with restricted centralizers of word-values. We show that if w is a multilinear commutator word and G a profinite group with restricted centralizers of w-values, then the verbal subgroup w(G) is abelian-by-finite.