Subgroups and Homology of Extensions of Centralizers of Pro-p Groups

TL;DR

This paper investigates the homological properties and structural characteristics of certain pro-p groups formed by extending centralizers of free pro-p groups, providing insights into their asymptotic homology dimensions.

Contribution

It introduces new results on the asymptotic behavior of homology groups and structural properties of pro-p groups arising from extensions of centralizers.

Findings

Homology dimensions grow asymptotically in a specific manner.

Structural properties of these pro-p groups are characterized.

New group-theoretic insights into extensions of centralizers are provided.

Abstract

We study asymptotic behavior of the dimensions of the homology groups of subgroups of finite index in finitely generated subgroups of pro-$p$ extension of centralizers of free pro-$p$ groups. We also prove group theoretic structure properties of such pro-$p$ groups that are of independent interest.