On lengths of HZ-localization towers

TL;DR

This paper investigates the $H ext{Z}$-length of groups, establishing bounds for free noncyclic groups and certain semi-direct products, and describes their $H ext{Z}$-localization structure.

Contribution

It provides new bounds for the $H ext{Z}$-length of specific groups and describes the structure of their $H ext{Z}$-localizations, especially for semi-direct products with cyclic modules.

Findings

Free noncyclic groups have $H ext{Z}$-length at least $oldsymbol{\omega+2}$.

Semi-direct products $M times C$ have $H ext{Z}$-length at most $oldsymbol{\omega+1}$.

$H ext{Z}$-localization of these groups is a central extension of their pro-nilpotent completion.

Abstract

In this paper, the $H\mathbb Z$-length of different groups is studied. By definition, this is the length of $H\mathbb Z$-localization tower or the length of transfinite lower central series of $H\mathbb Z$-localization. It is proved that, for a free noncyclic group, its $H\mathbb Z$-length is $\geq \omega+2$. For a large class of $\mathbb Z[C]$-modules $M,$ where $C$ is an infinite cyclic group, it is proved that the $H\mathbb Z$-length of the semi-direct product $M\rtimes C$ is $\leq \omega+1$ and its $H\mathbb Z$-localization can be described as a central extension of its pro-nilpotent completion. In particular, this class covers modules $M$, such that $M\rtimes C$ is finitely presented and $H_2(M\rtimes C)$ is finite.