Profinite groups in which the centralizer of any non-identity element  is abelian (i.e., profinite CA-groups) are also known as profinite  commutativity-transitivity groups.

In this talk I shall present a dichotomy theorem obtained with  P. Shumyatsky and P. Zalesskii (2019, Israel J. Math, v. 230):  Any profinite CA-group has a finite index closed subgroup that  is either abelian or pro-p.