Automorphisms and monomorphisms of direct products of virtually solvable minimax groups

TL;DR

This paper characterizes automorphisms and monomorphisms of direct products of virtually solvable minimax groups, extending known results from nilpotent groups and applying algebraic hull techniques.

Contribution

It provides a unique factorization of monomorphisms under an indecomposability assumption, extending the theory from nilpotent to virtually solvable minimax groups.

Findings

Every monomorphism factorizes uniquely into a permutation and a central off-diagonal component.

Automorphisms correspond exactly to permutations with isomorphic hulls.

The indecomposability assumption is proven to be sharp.

Abstract

This paper studies automorphisms and monomorphisms of direct products $\Gamma=\Gamma_1\times\cdots\times\Gamma_r$ of finitely generated virtually solvable minimax groups, a class containing all virtually polycyclic groups. Under an indecomposability assumption on the $\mathbb Q$-algebraic hulls, we prove that every monomorphism of $\Gamma$ factorizes uniquely as $\varphi=\theta\cdot\zeta$, where $\theta$ sends each factor into a permuted factor with $\mathbb Q$-isomorphic hull and $\zeta$ is central and off-diagonal. Conversely, every such pair defines a monomorphism of $\Gamma$, and $\varphi$ is an automorphism if and only if $\theta$ is. This indecomposability assumption is sharp: we show it cannot be weakened to direct indecomposability of the factors. The proof proceeds in three steps: first by establishing the corresponding central mixing property for finite-dimensional Lie algebras and algebraic Lie algebras, then for connected linear algebraic groups, and finally by transferring these results to minimax groups via $\mathbb Q$-algebraic hulls. This extends the previously known nilpotent case both from automorphisms to monomorphisms and from finitely generated torsion-free nilpotent groups to the broader class of finitely generated virtually solvable minimax groups. As applications, we characterize co-Hopfian direct products and derive formulas for Reidemeister numbers and Reidemeister spectra.