Automorphism groups of solvable groups of finite abelian ranks

TL;DR

This paper introduces a new explicit construction of the $ ext{Q}$-algebraic hull for virtually solvable groups with finite abelian ranks, analyzing their automorphism groups and applications to algebraic and topological conjectures.

Contribution

It provides a novel explicit construction of the $ ext{Q}$-algebraic hull for these groups and studies their automorphism groups in detail, including $S$-arithmetic properties.

Findings

Natural subgroups are $S$-arithmetic when $Fitt( ext{Gamma})$ is $S$-arithmetic.

$ ext{Out}( ext{Gamma})$ has an $S$-arithmetic image in algebraic outer automorphisms.

Applications towards Nekrashevych-Pete conjecture and fixed point theory.

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.