Let $X$ be an irreducible variety and $\operatorname{Bir}(X)$ its group of birational transformations. We show that the group structure of $\operatorname{Bir}(X)$ determines whether $X$ is rational and whether $X$ is ruled. Additionally, we prove that any Borel subgroup of $\operatorname{Bir}(X)$ has derived length at most twice the dimension of $X$ , with equality occurring if and only if $X$ is rational and the Borel subgroup is standard. We also provide examples of non-standard Borel subgroups of $\operatorname{Bir}(\mathbb{P}^{n})$ and $\operatorname{Aut}(\mathbb{A}^{n})$ , thereby resolving conjectures by Popov and Furter-Poloni.
