Rational cohomology and Zariski dense subgroups of solvable linear algebraic groups

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.