The cohomology of virtually torsion-free solvable groups of finite rank

TL;DR

This paper investigates the cohomology and homological properties of virtually torsion-free solvable groups of finite rank, establishing conditions for finiteness of cohomology groups and exploring structural group properties.

Contribution

It introduces a condition on modules ensuring finite cohomology and applies it to derive new results on near supplements, complements, and homological dimension in solvable groups.

Findings

Cohomology groups are finite under specific module conditions.

Derived new results on near supplements and complements in solvable groups.

Established a link between cohomology properties and homological dimension.

Abstract

Assume that $G$ is a virtually torsion-free solvable group of finite rank and $A$ a $\mathbb ZG$-module whose underlying abelian group is torsion-free and has finite rank. We stipulate a condition on $A$ that ensures that $H^n(G,A)$ and $H_n(G,A)$ are finite for all $n\geq 0$. Using this property for cohomology in dimension two, we deduce two results concerning the presence of near supplements and complements in solvable groups of finite rank. As an application of our near-supplement theorem, we obtain a new result regarding the homological dimension of solvable groups.