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Automorphism groups of solvable groups of finite abelian ranks

Jonas Deré, Mark Pengitore

Abstract

This paper gives a new explicit construction of the $\mathbb{Q}$-algebraic hull for virtually solvable groups $Γ$ of finite abelian ranks, taking into account the spectrum $S$ of the group $Γ$. As an application, we make a detailed study of the structure of $Aut(Γ)$ in the finitely generated case and show that a number of natural subgroups are $S$-arithmetic under the condition that $Fitt(Γ)$ is $S$-arithmetic. We then proceed by demonstrating that $Out(Γ)$ 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.

Automorphism groups of solvable groups of finite abelian ranks

Abstract

This paper gives a new explicit construction of the -algebraic hull for virtually solvable groups of finite abelian ranks, taking into account the spectrum of the group . As an application, we make a detailed study of the structure of in the finitely generated case and show that a number of natural subgroups are -arithmetic under the condition that is -arithmetic. We then proceed by demonstrating that has a -arithmetic image in the group of algebraic outer automorphisms of the -algebraic hull. We finish by discussing further applications of the -algebraic hull towards an open conjecture by Nekrashevych and Pete and topological fixed point theory.

Paper Structure

This paper contains 25 sections, 57 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Background: $\mathbb{Q}$-algebraic hulls
  3. Main results
  4. Open questions
  5. Plan of the paper
  6. Background
  7. Solvable groups of finite rank
  8. Linear algebraic groups
  9. Algebraic hulls for FAR-groups
  10. Construction of the $\mathbb{Q}$-algebraic hull
  11. Properties of the $\mathbb{Q}$-algebraic hull
  12. Thickenings and virtually nilpotent supplements
  13. The automorphism group of the thickening
  14. Thickenings of $\Gamma$ admit supplements
  15. Unipotent shadows of solvable groups of finite abelian ranks
  16. ...and 10 more sections

Key Result

Theorem A

A virtually torsion-free solvable group of finite abelian ranks $\Gamma$ with spectrum $S$ has a $\mathbb{Q}$-algebraic hull if and only if $\Gamma$ has a trivial maximal normal torsion subgroup. Moreover, the $\mathbb{Q}$-algebraic hull is unique up to algebraic isomorphism, and every monomorphism

Theorems & Definitions (114)

  • Definition 1.1
  • Definition 1.2
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 104 more