Compact $p$-adic analytic groups in which centralizers are abelian

TL;DR

This paper constructs specific profinite $p$-adic analytic groups with the property that all non-trivial element centralizers are abelian, addressing open questions in the field.

Contribution

It introduces a novel construction of $p$-adic analytic groups with abelian centralizers, using advanced algebraic and cohomological methods.

Findings

Constructed examples of $p$-adic analytic groups with abelian centralizers.

Provided answers to open questions posed by previous researchers.

Applied diverse mathematical techniques including Lie theory and modular representation theory.

Abstract

Using methods of associative algebras, Lie theory, group cohomology, and modular representation theory, we construct profinite $p$-adic analytic groups such that the centralizer of each of their non-trivial elements is abelian. The paper answers questions of P.~Shumyatsky, P.~Zalesskii, and T.~Zapata in the Israel J. Math., v.~230, 2019.