The torsion-free rank of homology in towers of soluble pro-p groups

TL;DR

This paper establishes an upper bound on the torsion-free rank of the first homology group for certain finitely presented pro-p groups, revealing structural constraints in their subgroup towers.

Contribution

It introduces a bound on the homology torsion-free rank in towers of finitely presented pro-p nilpotent-by-abelian-by-finite groups, a novel insight into their algebraic structure.

Findings

Bound on the dimension of homology groups for finite index subgroups

Structural constraints on pro-p nilpotent-by-abelian-by-finite groups

Advancement in understanding subgroup homology in pro-p groups

Abstract

We show that for every finitely presented pro-$p$ nilpotent-by-abelian-by-finite group $G$ there is an upper bound on $\dim_{\mathbb{Q}_p} (H_1(M, \mathbb{Z}_p) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p )$, as $M$ runs through all pro-$p$ subgroups of finite index in $G$.