Solvable Automorphism Groups of Varieties

Let $X$ be a variety of dimension $n$, and let $\mathrm{A}\mathrm{u}\mathrm{t}(X)$ be its automorphism group. When $X$ is quasi-affine, we prove that a solvable subgroup of $\mathrm{A}\mathrm{u}\mathrm{t}(X)$ that is generated by an irreducible family of automorphisms containing the identity is an algebraic subgroup. Our main applications concern arbitrary varieties. First, every connected solvable subgroup of $\mathrm{A}\mathrm{u}\mathrm{t}(X)$ is contained in a Borel subgroup and its derived length is $\le n+1$. Second, the notion of solvable and unipotent radicals are well defined for any subgroup of $\mathrm{A}\mathrm{u}\mathrm{t}(X)$. Third, if $X$ is quasi-affine and connected and $\mathcal{B}\subset \mathrm{A}\mathrm{u}\mathrm{t}(X)$ is a Borel subgroup of derived length $n+1$, then $X$ is isomorphic to the affine $n$-space ${\mathbb{A}}^n$ and $\mathcal{B}$ is conjugate to the Jonquières subgroup.