Torsion cohomology for solvable groups of finite rank

TL;DR

This paper investigates the torsion cohomology of solvable groups of finite rank, establishing conditions under which their cohomology groups are finite or Cernikov groups, extending understanding of their algebraic structure.

Contribution

It introduces a class of solvable groups of finite rank and provides new criteria for the finiteness and Cernikov properties of their cohomology groups.

Findings

Conditions for $H^n(G,A)$ and $H_n(G,A)$ to be finite for certain modules.

Criteria for $H^n(G,A)$ to be a Cernikov group when $A$ is a Cernikov group.

Extension of cohomological understanding for solvable groups of finite rank.

Abstract

We define a class $\mathcal{U}$ of solvable groups of finite abelian section rank which includes all such groups that are virtually torsion-free as well as those that are finitely generated. Assume that $G$ is a group in $\mathcal{U}$ and $A$ a $\mathbb ZG$-module. If $A$ is $\mathbb Z$-torsion-free and has finite $\mathbb Z$-rank, we stipulate a condition on $A$ that guarantees that $H^n(G,A)$ and $H_n(G,A)$ must be finite for $n\geq 0$. Moreover, if the underlying abelian group of $A$ is a \v{C}ernikov group, we identify a similar condition on $A$ that ensures that $H^n(G,A)$ must be a \v{C}ernikov group for all $n\geq 0$.