Automorphisms and monomorphisms of direct products of virtually solvable minimax groups

Abstract

This paper studies automorphisms and monomorphisms of direct products $Γ=Γ_1\times\cdots\timesΓ_r$ 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 $Γ$ factorizes uniquely as $φ=θ\cdotζ$, where $θ$ sends each factor into a permuted factor with $\mathbb Q$-isomorphic hull and $ζ$ is central and off-diagonal. Conversely, every such pair defines a monomorphism of $Γ$, and $φ$ is an automorphism if and only if $θ$ 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.