All Questions

Let $\Gamma$ be a finitely generated finitely presented virtually solvable group. Assume that there exists an injective representation $\Gamma \to \operatorname{GL}_n(\mathbb{C})$. Is it true that ...

Is every connected solvable algebraic group a Borel subgroup of a reductive group? If a counterexample exists, I would ideally like it to be over $\Bbb C$.

Let $K$ be a (locally compact) local field and $G$ be a linear algebraic $K$-group. Does the topological group $G(K)$ have a cocompact solvable closed subgroup? If $\mathrm{char}(K)=0$, it is true ...

The basic question is whether there is a notion of chief factor of a connected solvable algebraic group that matches my intuition. A few smaller assertions are sprinkled through the explanation, and ...