Central theorems for cohomologies of certain solvable groups

TL;DR

This paper establishes that the cohomology of certain solvable groups can be computed via algebraic group rational cohomology, generalizing previous results and providing new proofs for known theorems.

Contribution

It introduces a unifying framework linking group cohomology of solvable groups with algebraic group rational cohomology, extending prior work on solvmanifolds.

Findings

Cohomology of torsion-free virtually polycyclic groups is computable via algebraic groups.

Continuous cohomology of simply connected solvable Lie groups aligns with algebraic group rational cohomology.

Provides a simplified proof of the Dekimpe-Igodt cohomology vanishing theorem.

Abstract

We show that the group cohomology of torsion-free virtually polycyclic groups and the continuous cohomology of simply connected solvable Lie groups can be computed by the rational cohomology of algebraic groups. Our results are generalizations of certian results on the cohomology of solvmanifolds and infra-solvmanifolds. Moreover as an application of our results, we give a simple proof of the surprising cohomology vanishing theorem given by Dekimpe-Igodt.