Automorphism groups of solvable groups of finite abelian ranks

Abstract: The $\mathbb{Q}$-algebraic hull of a virtually polycyclic group $\Gamma$ is an important tool to study its group of automorphisms $\Aut(\Gamma)$ using linear algebraic groups. For example, it was used by Baues and Gr\"unewald to prove that $\Out(\Gamma)$ is an arithmetic group, although $\Aut(\Gamma)$ is not for a general virtually polycyclic group. The original definition due to Mostow was generalized to the class of virtually solvable groups of finite abelian ranks by Arapura and Nori to study K\"ahler manifolds, although other applications were not considered. This paper gives a new explicit construction of the $\Q$-algebraic hull for virtually solvable groups $\Gamma$ of finite abelian ranks, taking into account the spectrum $S$ of the group $\Gamma$. As an application, we make a detailed study into the structure of $\Aut(\Gamma)$ and show that several natural subgroups are $S$-arithmetic under the condition that $\Fitt(\Gamma)$ is $S$-arithmetic. Additionally, we demonstrate that the natural generalization of the aforementioned result by Baues and Gr\"unewald fails even for most solvable Baumslag-Solitar groups. Among the other applications of our main result is also a theorem showing that if $\Gamma$ is strongly scale-invariant, it must be virtually nilpotent, giving a special case of a conjecture by Nekrashevych and Pete and an extension of the previous known result for virtually polycyclic groups.