Automorphism groups of solvable groups of finite abelian ranks

2025/06/09 by Jonas Deré, Deré, Jonas, Mark Pengitore +1
Mathematics · Computer Science · #Finite Group Theory Research #Polynomial and algebraic computation #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2506.07991

Abstract

This paper gives a new explicit construction of the 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 ℚ-algebraic hull. We finish by discussing further applications of the ℚ-algebraic hull towards an open conjecture by Nekrashevych and Pete and topological fixed point theory.

Citations

• Strongly scale-invariant virtually polycyclic groups
• Solvable fundamental groups of algebraic varieties and Kähler manifolds
• Dynamical Zeta Functions, Nielsen theory and Reidemeister torsion

Related

• Solvable subgroups of Out(F_n) are virtually abelian
• Nonarchimedean superrigidity of solvable S-arithmetic groups
• The automorphism groups of relatively free groups of infinite rank
• A finitely-presented solvable group with a small quasi-isometry group
• Solvable fundamental groups of algebraic varieties and Kähler manifolds