Automorphism groups of solvable groups of finite abelian ranks

Abstract

This paper gives a new explicit construction of the $\mathbb{Q}$-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 of the structure of $Aut(\Gamma)$ in the finitely generated case and show that a number of natural subgroups are $S$-arithmetic under the condition that $Fitt(\Gamma)$ is $S$-arithmetic. We then proceed by demonstrating that $Out(\Gamma)$ has a $S$-arithmetic image in the group of algebraic outer automorphisms of the $\mathbb{Q}$-algebraic hull. We finish by discussing further applications of the $\mathbb{Q}$-algebraic hull towards an open conjecture by Nekrashevych and Pete and topological fixed point theory.