Automorphisms and monomorphisms of direct products of virtually solvable minimax groups

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 $\phi =\theta \cdot \zeta$, 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 $\phi$ 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.

v2: 30 pages. v3: 31 pages; minor revisions incorporating the referees' comments. Published in Transformation Groups