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Homological growth of nilpotent-by-abelian pro-p groups

Dessislava H. Kochloukova, Aline G. S. Pinto

TL;DR

This work analyzes the growth of homology in pro-$p$ groups that are nilpotent-by-abelian, focusing on how the torsion-free rank of $H_i(M,\mathbb{Z}_p)$ behaves when $M$ ranges over finite-index pro-$p$ subgroups of a fixed group $G$. The authors introduce $T$-maps to compare homology across finite-index subgroups and leverage Lyndon–Hochschild–Serre spectral sequences to relate $H_j(N,\mathbb{Z}_p)$ to completed tensor powers of $N/N'$, controlled by the diagonal action of the abelian quotient $Q$. Under the hypothesis that the metabelian quotient $G/N'$ is of type $FP_{2d}$ with $d=cm$ (where $N$ has nilpotency class $c$ and $Q$ is abelian), they prove that the torsion-free rank of $H_i(M,\mathbb{Z}_p)$ remains uniformly bounded for $0\le i\le m$, as $M$ varies over finite-index subgroups. This extends prior results for more restricted soluble pro-$p$ groups and builds a framework connecting $FP_{k}$ conditions to homology growth via filtrations built from completed tensor products and exterior powers, exposing a structural bridge between finiteness properties and homological growth in pro-$p$ groups.

Abstract

We show that the torsion-free rank of $H_i(M, \mathbb{Z}_p)$ has finite upper bound for $i \leq m$, where $M$ runs through the pro-$p$ subgroups of finite index in a pro-$p$ group $G$ that is (nilpotent of class $c$)-by-abelian such that $ G/N'$ is of type $FP_{2cm}$.

Homological growth of nilpotent-by-abelian pro-p groups

TL;DR

This work analyzes the growth of homology in pro- groups that are nilpotent-by-abelian, focusing on how the torsion-free rank of behaves when ranges over finite-index pro- subgroups of a fixed group . The authors introduce -maps to compare homology across finite-index subgroups and leverage Lyndon–Hochschild–Serre spectral sequences to relate to completed tensor powers of , controlled by the diagonal action of the abelian quotient . Under the hypothesis that the metabelian quotient is of type with (where has nilpotency class and is abelian), they prove that the torsion-free rank of remains uniformly bounded for , as varies over finite-index subgroups. This extends prior results for more restricted soluble pro- groups and builds a framework connecting conditions to homology growth via filtrations built from completed tensor products and exterior powers, exposing a structural bridge between finiteness properties and homological growth in pro- groups.

Abstract

We show that the torsion-free rank of has finite upper bound for , where runs through the pro- subgroups of finite index in a pro- group that is (nilpotent of class )-by-abelian such that is of type .

Paper Structure

This paper contains 6 sections, 12 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. T-maps
  4. Proof of Theorem \ref{['principal']}
  5. Homology of nilpotent pro-$p$ subgroups
  6. Coming back to the proof of Theorem \ref{['principal']}

Key Result

Theorem 1.1

Let $1\to N \to G \to Q \to 1$ be an short exact sequence of pro-$p$ groups, where $G$ is finitely generated, $N$ is nilpotent of class $c$ and $Q$ is abelian. Let $N'$ be the commutator subgroup of $N$ and suppose that the metabelian quotient $G/N'$ of $G$ is of type $FP_{2d}$, where $d = cm$. Then where $\mathcal{A}$ is the set of all subgroups of $p$-power index in $G$ and, for an abelian pro-$

Theorems & Definitions (18)

  • Theorem 1.1
  • Lemma 2.1: KP-centre-by-metabelian
  • Lemma 2.2: KP-centre-by-metabelian
  • Lemma 2.3: KP-centre-by-metabelian
  • Theorem 2.4: KP-centre-by-metabelian
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 8 more