Rational cohomology and Zariski dense subgroups of solvable linear algebraic groups

TL;DR

This paper investigates the cohomology of Zariski dense subgroups within solvable linear algebraic groups, establishing isomorphisms and injectivity results for cohomology rings and restriction maps.

Contribution

It proves isomorphisms between cohomology rings of algebraic groups and their Zariski dense subgroups, and shows the restriction map in rational cohomology is injective under certain conditions.

Findings

Cohomology rings of algebraic groups and Zariski dense subgroups are isomorphic.

Restriction map in rational cohomology from the group to Zariski dense subgroup is injective.

Results on finitely generated solvable groups of finite abelian rank and their cohomology representations.

Abstract

In this article, we establish results concerning the cohomology of Zariski dense subgroups of solvable linear algebraic groups. We show that for an irreducible solvable $\mathbb{Q}$-defined linear algebraic group $\mathbf{G}$, there exists an isomorphism between the cohomology rings with coefficients in a finite dimensional rational $\mathbf{G}$-module $M$ of the associated $\mathbb{Q}$-defined Lie algebra $\mathfrak{g_\mathbb{Q}}$ and Zariski dense subgroups $\Gamma \leq \mathbf{G}(\mathbb{Q})$ that satisfy the condition that they intersect the $\mathbb{Q}$-split maximal torus discretely. We further prove that the restriction map in rational cohomology from $\mathbf{G}$ to a Zariski dense subgroup $\Gamma \leq \mathbf{G}(\mathbb{Q})$ with coefficients in $M$ is an injection. We then derive several results regarding finitely generated solvable groups of finite abelian rank and their representations on cohomology.